Abstract Here we determine up to conjugacy all the maximal subgroups of the finite exceptional group of Lie-type E7(2). 370 Introduction Interest in maximal subgroups of finite simple groups can be traced back to the dawn of group theory, in the guise of primitive permutation groups. The letter Galois wrote to Chevallier [19] on May 29, 1832 , the evening before the fatal duel, which includes statements on the minimal (non-trivial) permutation degrees of L 2 (p), p a prime, is one such example. Later Dickson [17] in 1901 determined all the subgroups of L 2 (q) (q a power of a prime), while those for L 3 (q) were similarly classified by Mitchell [36] in 1911 (for q odd) and Hartley [22] in 1925 (for q even). Introduction As the work on the finite simple groups, which would ultimately lead to their classification, gathered pace so too did investigations into the maximal subgroups of the simple groups. For many of the sporadic groups machine calculations figure prominently; at the moment all maximal subgroups of the sporadic groups, apart from those of the Monster, are known. Not surprisingly, there is still much to be learnt about the maximal subgroups of the simple groups of Lie-type. This area has been the subject of intense investigation by numerous authors over the last sixty years. The approaches used have been wide and diverse, ranging from exploiting certain multilinear forms associated with the groups [2, 3, 15] , through the theory of linear algebraic groups [30, 33] , to a mixture of both theoretical and computational work [24, 37] . Among the many milestones we mention Aschbacher [4] , who introduced the Aschbacher classes for the classical groups, and the recent work by Liebeck, Saxl and Seitz amongst others on the maximal subgroups of exceptional algebraic and finite groups of Lie-type. This latter body of work forms the starting point for our work here, so we summarise this as Theorem 2.1 in our next section. Our interest here is in the exceptional group of Lie-type E 7 (2); for the state of play with the other exceptional groups see [39, Chapter 4] and references therein. The main result of this paper is as follows (notation will be discussed later in this section). (2)). Introduction Furthermore, maximal subgroups of each shape above exist in E 7 (2) , and these maximal subgroups are determined uniquely up to conjugacy. Introduction We shall use Sym(n), respectively Alt(n), to denote the symmetric group, respectively alternating group, of degree n. Also Ω ± n (q) will stand for the simple projective orthogonal group of either plus type or minus type, Sp n (q) for the symplectic groups and Dih(m) for the dihedral group of order m. Apart from these exceptions, our notation for group structures follows the Atlas [14] . Moreover the Atlas will be a significant source for information on many of the groups we encounter. For general group theoretic notation see either [1] or [21] , and we use the symbol ∼ to indicate that two groups have the same shape. Introduction We now give an indication of the strategy and methods used to prove Theorem 1.1, as well as the layout of the paper. We shall see that, as a consequence of Theorem 2.1, to prove Theorem 1.1 there is a finite list (see List 1) of groups for which we must determine whether or not they are maximal in E 7 (2) . As it turns out, only one group, up to conjugacy in E 7 (2), arising from this list is in fact maximal in E 7 (2) . Section 3 is devoted to assembling an extensive catalogue of information concerning the involutions and semisimple elements of E 7 (2) . This data is used in a variety of ways, such as for example in Lemma 4.2, to demonstrate that L 2 (64), Sp 4 (8), L 4 (8) and Sp 6 (8) cannot be subgroups of E 7 (2), while Lemma 3.6, used in Lemma 4.21 to eliminate H being a maximal subgroup with Soc(H) ∼ = L 4 (4), gives us an Ω − 8 (2) subgroup of E 7 (2) within which we can work. The 56-dimensional irreducible F 2 -module for E 7 (2) is used at almost every turn in our analysis, either for computational attacks or in considering the possible restrictions for subgroups of E 7 (2) upon this module. Use of Brauer characters will be seen to be a potent weapon for either eliminating certain possible subgroups of E 7 (2) (for example see Lemmas 4.3 and 5.2) or in pinpointing those cases needing further investigation. Thus Appendix A contains a large number of (partial) Brauer character tables (with each irreducible Brauer character labelled by χ i for some i) while § § 4 and 5 are where we consider, respectively, those possible maximal subgroups H where Soc(H) ∈ Lie(2), the Lie-type groups of characteristic 2, and Soc(H) / ∈ Lie(2). For those cases when H is a possible maximal subgroup of E 7 (2) with Soc(H) being isomorphic to a small simple group such as L 3 (2) or L 2 (8) , their elimination is a protracted campaign involving a mainly computational approach. For the former case this is played out in Lemmas 4.13-4.19. The latter case is dealt with in Lemma 4.20 and also makes heavy use of facts about normalisers of cyclic subgroups of order 7 given in § 4.2. As a further comment, we note that the method in [7] for finding (computationally) generators for centralisers of strongly real elements is used extensively. Introduction Where extensive computation has been employed, we have provided downloadable data files which document these computations (available as online supplementary material from the publisher's website). Names of specific files associated to a particular result are given at the beginning of the relevant proof. Introduction Finally, we would like to thank the referee, whose comments and suggestions have greatly improved this paper. Preliminary results The first theorem of this section, already mentioned in the previous section, summarises the work of a variety of authors in [25] [26] [27] [28] [29] [31] [32] [33] and [38] . For Theorem 2.1, G denotes an adjoint simple algebraic exceptional group of Lie-type over F q and σ is a standard Frobenius homomorphism of G. Preliminary results Theorem 2.1. Let H be a maximal subgroup of the finite exceptional group G σ over F q , q = p a where p is a prime. Then one of the following holds: Table 1 ], and H = M σ as in [30, Table 3 ]; (ii) H is of the same type as G; (iii) H is an exotic local subgroup (see [33] ); (iv) G is of type E 8 , p > 5 and H ∼ (Alt(5) × Alt(6)).2 2 ; Preliminary results (v) F * (H) = H 0 is simple, and not in Lie(p): the possibilities for H 0 are given up to isomorphism by [31] ; (vi) F * (H) = H(q 0 ) is simple and in Lie(p); moreover rk(H(q 0 )) In cases (i)-(iv), H is determined up to G σ -conjugacy. Preliminary results For the remainder of this paper, G will denote E 7 (2) and V its (minimal) 56-dimensional F 2 G-module. Thus, from [14] , |G| = 2 63 .3 11 .5 2 .7 Preliminary results Let H be a maximal subgroup of G. In view of Theorem 2.1 and calling on the finite simple group classification, to prove Theorem 1.1 we must consider the following possibilities. List 1. (i) F * (H) is isomorphic to one of the following groups: L4 (2) L4 (4) L4 (8) U4 (2) U4 (4) U4 (8) Sp6 (2) Sp6 (4) Sp6 (8) L3 (2) L3 (4) L3 (8) U3 (4) U3 (8) Sp4 (4) Sp4 (8) G2 (4) G2(8) Sz (8) L2 (8) . List 1. (i) F * (H) is isomorphic to one of the following groups: These groups arise from Theorem 2.1(vi)(a). (ii) F * (H) is isomorphic to L 3 (16) or U 3 (16)(these possibilities arise from Theorem 2.1(vi)(b)). (32) or Sz(128) (these possibilities arise from Theorem 2.1(vi)(c)). (iv) F * (H) is isomorphic to one of the following groups: List 1. (i) F * (H) is isomorphic to one of the following groups: (a) Alt(n) for n = 5, 6, 7, 8, 9, 10, 11, 12 and 13; (b) L 2 (q) for q = 7, 11, 13, 17, 19, 25, 27, 29 and 37; List 1. (i) F * (H) is isomorphic to one of the following groups: These possibilities arise from Theorem 2.1(v). List 1. (i) F * (H) is isomorphic to one of the following groups: When analysing potential subgroups H of G, we shall often show the non-maximality of H by demonstrating that H must fix a non-zero vector in V . The next four results are used in this regard. Proposition 2.2. The vector stabilisers of G on V \{0} have the following shapes: (i) 2 List 1. (i) F * (H) is isomorphic to one of the following groups: 27 : E 6 (2); List 1. (i) F * (H) is isomorphic to one of the following groups: (ii) E 6 (2) : 2; List 1. (i) F * (H) is isomorphic to one of the following groups: Proof. The point-stabilisers of E 7 (q) (for arbitrary q) in its action on its minimal 56-dimensional module are determined in [26] . In order to obtain the structures given in the statement of the proposition for the special case where q = 2 we can use information from [5] . List 1. (i) F * (H) is isomorphic to one of the following groups: the maximal subgroups of E 7 (2) must hold: List 1. (i) F * (H) is isomorphic to one of the following groups: (ii) N G (S) is contained in one of the vector stabilisers described in Lemma 2.2. List 1. (i) F * (H) is isomorphic to one of the following groups: Proof. Assume that C G (S) = 1. Then since the automiser N G (S)/C G (S) is a subgroup of Aut(S) we know that N G (S) embeds in Aut(S). Further, since S is simple we have Z(S) = 1, and so Inn(S) ∼ = S/Z(S) ∼ = S. Thus Out(S) = Aut(S)/Inn(S) ∼ = Aut(S)/S and so N G (S)/S is a 2-group. Since S fixes a non-zero vector in V and N G (S)/S acts on this subspace we know that N G (S) fixes a non-zero vector in V by Lemma 2.3. Hence N G (S) must lie in a vector stabiliser as claimed. List 1. (i) F * (H) is isomorphic to one of the following groups: The following result allows us to use knowledge of the dimensions of certain cohomology groups to deduce information regarding the action of a potential subgroup H G on V . List 1. (i) F * (H) is isomorphic to one of the following groups: Proposition 2.5. Suppose that S is a finite group, with k a field of characteristic p > 0. Let W be a finite dimensional kS-module, with V the projective indecomposable module corresponding to W . Then the number of trivial modules in an S-composition series of List 1. (i) F * (H) is isomorphic to one of the following groups: Proof. See [8] . List 1. (i) F * (H) is isomorphic to one of the following groups: We conclude this section with a brief description of the construction of E 7 (2) which we use for the majority of our computations. Not surprisingly the root system of type E 7 will feature in this construction and at various points in our arguments. So we first set up the labelling of the E 7 Dynkin diagram as follows. List 1. (i) F * (H) is isomorphic to one of the following groups: We shall employ the ordering of the positive roots as given in Table 1 . This ordering is first by height and then by lexicographic order with respect to the labelling of the fundamental roots. Notationally α i will denote the ith root in this ordering system. Also we set α 0 := α 63 , the highest root in the root system. For 64 i 126 we then set α i to be the negative of α i−63 . List 1. (i) F * (H) is isomorphic to one of the following groups: Following the construction of Chevalley groups over arbitrary fields as described in [12] , we first produce generators for E 8 (2) acting on its 248-dimensional adjoint module V 248 (with respect to a Chevalley basis). These generators correspond to roots from the root system of type E 8 . We now use the fact that E 7 (2) is a subgroup of E 8 (2) (see [39] , for example). Taking the canonical subgroup E 7 (2) which corresponds to the first seven nodes of the E 8 Dynkin diagram, we use the implementation of Parker's Meataxe in Magma [10] to decompose V 248 as an E 7 (2)-module. The minimal 56-dimensional module for E 7 (2) appears as an E 7 (2)-submodule of V 248 , while the irreducible 132-dimensional module, which we denote V 132 , appears as a composition factor of V 248 . It is now straightforward to map our canonical root element generators to those for the 56-and 132-dimensional representations of E 7 (2). List 1. (i) F * (H) is isomorphic to one of the following groups: (1 2 2 3 2 1 1) α57 List 1. (i) F * (H) is isomorphic to one of the following groups: (1 1 2 3 2 2 1) α58 List 1. (i) F * (H) is isomorphic to one of the following groups: (1 2 2 3 2 2 1) α59 List 1. (i) F * (H) is isomorphic to one of the following groups: (1 1 2 3 3 2 1) α60 List 1. (i) F * (H) is isomorphic to one of the following groups: (1 2 2 3 3 2 1) α61 List 1. (i) F * (H) is isomorphic to one of the following groups: (1 2 2 4 3 2 1) α62 List 1. (i) F * (H) is isomorphic to one of the following groups: (1 2 3 4 3 2 1) α63 = α0 (2 2 3 4 3 2 1) Involutions and semisimple elements Involutions in G The following result is found in [5] , and gives information on the G-conjugacy of involutions. Involutions in G Theorem 3.1. Each involution in G = E 7 (2) is conjugate to one of the following: Involutions in G Proof. See [5, 16.1] , using the fact that α 0 = r 27 , α 59 = r 21 , α 58 = r 23 , α 53 = r 48 , α 55 = r 18 , α 54 = r 19 , α 48 = r 13 , α 47 = r 14 and α 49 = r 15 (r j as defined in [5] ). Involutions in G Let P i denote the maximal parabolic subgroup of G obtained by the removal of the ith node from the Dynkin diagram. Also we use Q i to denote the maximal normal unipotent subgroup of P i , and L i its Levi complement. Involutions in G Theorem 3.2. The maximal parabolic subgroups containing the centralisers of the involutions of G are as follows: Involutions in G Proof. See [5, 16.20] . Involutions in G Theorem 3.3. For t an involution of G, let U denote the maximal normal unipotent subgroup of C G (t). The possible structures of C G (t) are as follows: Involutions in G is an extraspecial 2-group and so Q 1 /Z(Q 1 ) has the structure of an orthogonal space, upon which Involutions in G Proof. See [5, 16.20] . Involutions in G In accordance with the Atlas [14] convention of largest centralisers having precedence, we label the t 1 -t 5 G-conjugacy classes by, respectively, 2A, 2B, 2D, 2C and 2E. Involutions in G Lemma 3.4. Let t be an involution in the classes 2A, 2B, 2C, 2D or 2E of G. Then the rank of 1 + t on the module V is 12, 20, 24, 28 or 28, respectively. Involutions in G Proof. Firstly let us take the subgroup Involutions in G as the structure of involutions in the linear groups is well known. Now using the Meataxe we know that as an L-module we have Involutions in G * , where V 8 denotes the natural 8-dimensional module for L 8 (2) in characteristic 2 and Λ 2 (V 8 ) denotes its exterior square. Let t be an involution of L 8 (2) with one Jordan block of size 2 and six Jordan blocks of size 1 on V 8 . Now t is centralised by an L 6 (2) in L and so must be a 2A-element since no other involution is centralised by an element of order 31. Now such an involution has six Jordan blocks when represented on both Λ 2 (V 8 ) and Λ 2 (V 8 ) * and hence the rank of 1 + t is 12. Let s be an involution of L 8 (2) with two Jordan blocks of size 2 and four Jordan blocks of size 1 on V 8 . The element s is centralised by an Alt(8) × Sym(3) in L and so using [14] we see that it must be a 2B-element. This involution has ten Jordan blocks when represented on both Λ 2 (V 8 ) and Λ 2 (V 8 ) * , and hence the rank of 1 + s is 20. Involutions in G Now we take a non-diagonal involution r in N G (7B) ∼ (7 : 2 × 3 D 4 (2)) : 3 (see § 4.2) which inverts the 7B element (that is an involution in the dihedral group 7 : 2). Since r centralises an element of order 13 it must be a 2C-element, just by consideration of the centraliser orders. We find that dim(C V (r)) = 28 and the rank of 1 + r is 28. Now from [24] we know there is an involution v in E 6 (2) whose centraliser contains Sym(3) × L 3 (2) and for which 1 + v has rank 12 on the minimal E 6 (2)-module V 27 . Now as an E 6 (2)-module we have V = 1 ⊕ 1 ⊕ V 27 ⊕ V * 27 , and so on V we see that the rank of 1 + v is 24. This differs from the ranks so far accounted for and so the corresponding involution v in G is either in 2D or 2E. However, Sp 6 (2) contains no subgroups Sym(3) × L 3 (2) and so v must be in the class 2D. Finally we need to find the rank of a 2E-involution. For this we turn to the 56-dimensional matrix representation of G over F 2 . We take a 2E-involution u as described in [5] and find dim(C V (t 4 )) = 28. Thus 1 + u has rank 28 and we are done. Semisimple classes In preparation for the proof of Theorem 1.1, we require information on the structure of centralisers of semisimple elements of E 7 (2). Frank Lübeck has produced a parametrization of the conjugacy classes of E 7 (2), which is stored at [34] . It is well known that there are 128 conjugacy classes of semisimple elements in E 7 (2), and [34] gives the orders and structural information of the centralisers of such elements. In this section we determine the number of conjugacy classes of elements of each odd order appearing in E 7 (2), and write down the structure of each centraliser in the detail we require for later calculations. The results are recorded in Table 2 . We also record, in Tables 3 and 4 respectively, the Brauer character Table 2 . (Continued). Semisimple classes It is clear that to find the G-conjugacy classes of elements g of order pq in G (where (p, q) = 1) we need to determine the C G (g p )-classes of elements of order p. Lemma 3.7. There are seven classes of elements of order 15 in G, with centralisers as in Table 2 . Furthermore the fifth powers of elements from these classes lie in the following classes of G. Semisimple classes Proof. Any element of order 15 in G must cube to an element in 5A, since this is the unique class of elements of order 5 in G. By Lemma 3.6 we have C G (5A) ∼ = 5 × Ω − 8 (2) × Sym(3). Now from [14] we know that Ω − 8 (2) has three classes of elements of order 3, and we can deduce the structure of their centralisers. There is a class 3A with centraliser 3 × Alt(8), a class 3B with centraliser 3 × Alt(5) × Sym(3) and a class 3C with soluble centraliser 3 1+2 + : 2.Alt(4). Also, there is clearly a unique class of elements of order 3 in Sym(3) and such elements are self-centralising. Hence we find there are seven classes of elements of order 15 in G, with centralisers as in the statement of the lemma. The G-conjugacy classes in which the fifth powers of elements x lie were determined computationally. Semisimple classes Lemma 3.8. There are two classes of elements of order 45 in G, with centralisers as in Table 2 . In both cases the fifteenth power of such an element lies in the class 3E of G. Semisimple classes Proof. As in Lemma 3.7, any element of order 45 in G must power down to a 5A-element. Again by Lemma 3.6, (3) and from [14] we see that Ω − 8 (2) has a unique class of elements of order 9, and these are self-centralising. Hence there are two classes of elements of order 45 in G, namely 45A with centraliser of order 2. . Hence, from Lemma 3.7 we deduce that the elements in 45A and 45B both have fifteenth power in 3E of G. Semisimple classes Lemma 3.9. There is a unique class of elements of order 35 in G, with centraliser as in Table 2 . These elements power to 7A-elements in G. Semisimple classes Proof. From [14] we see that Ω − 8 (2) has a unique class of elements of order 7 with cyclic centraliser of order 21. This yields a unique class 35A in G with centraliser 35 × 3 × Sym(3). Hence, any element of order 35 in G lies in this class, and upon finding such an element g ∈ G and calculating dim C V (g), we find that the fifth power of g lies in 7A. Semisimple classes Lemma 3.10. There are five classes of elements of order 105 in G, with centralisers as in Table 2 . Semisimple classes Proof. By Lemma 3.9 any element of order 105 in G cubes to an element in 35A, and Semisimple classes Hence we obtain five classes of elements of order 105: two classes with centraliser 35 × 3 × Sym(3) corresponding to elements (35A, 3A, 1A) in C G (35A), two classes with centraliser 35 × 3 × 3 corresponding to elements (35A, 3A, 3A) in C G (35A), and one class with centraliser 35 × 3 × 3 corresponding to elements (35A, 1A, 3A) in C G (35A). Semisimple classes Lemma 3.11. There are four classes of elements of order 85 in G, with centralisers as in Table 2 . Semisimple classes Proof. From [14] we see that Ω − 8 (2) has four classes of elements of order 17, all of which are self-centralising. Using notation from [14] , these form a master class and subsequent slave classes, 17A, 17B 2, 17C 3 and 17D 6, which immediately yield G-conjugacy classes 85A, 85B 2, 85C 3 and 85D 6, all of which have centraliser of the form 85 × Sym(3). Semisimple classes Lemma 3.12. There are four classes of elements of order 255 in G, with centralisers as in Table 2 . Semisimple classes Proof. Any element of order 255 must cube to an 85-element. Using Lemma 3.11 we see that each class of elements of order 85 in G yield a self-centralising class of elements of order 255, and so we have 255A, 255B 2, 255C 3 and 255D 6. Lemma 3.13. There are three classes of elements of order 7 in G, with centralisers as in Table 2 . Semisimple classes Proof. Since |G| 7 = 7 3 and |E 6 (2)| 7 = 7 3 , the subgroup E 6 (2) contains a full Sylow 7-subgroup of G. From [24] we know that S ∈ Syl 7 (E 6 (2)) is elementary abelian of order 7 Semisimple classes 3 . There are four classes of elements of order 7 in E 6 (2), namely 7AB with centraliser 7 Semisimple classes . On the minimal module V 27 for E 6 (2) we have dim C V27 (7AB) = 0, dim C V27 (7C) = 9 and dim C V27 (7D) = 3. Now as an E 6 (2)-module we know that V = 1 ⊕ 1 ⊕ V 27 ⊕ V * 27 . Hence these elements in G have fixed spaces of dimensions 2, 2, 20 and 8 on V for elements in 7A, 7B, 7C and 7D, respectively. Hence the only possible fusion is between 7A and 7B classes in E 6 (2). Indeed these classes fuse under the graph automorphism of E 6 (2), and since E 6 (2) : 2 G we have three classes 7A, 7B and 7C in G. The structure of their centralisers follows from [18] . Semisimple classes Lemma 3.14. There are eight classes of elements of order 21 in G, with centralisers as in Table 2 . Semisimple classes Proof. Consider first those 21-elements whose cube is a 7A-element. Now Semisimple classes and L 6 (2) has three classes of elements of order 3. There is a class 3A with centraliser 3.L 3 (4).3, a class 3B with centraliser 3 × Alt (8) , and a class 3C with centraliser 3 × Alt(5) × Sym(3). Each of these yields a corresponding class of elements of order 21 in G. Now 7B-elements in G have centraliser 7 × 3 D 4 (2), and from [14] we see that 3 D 4 (2) has two classes of elements of order 3. These are a class 3A with centraliser 3 × L 2 (8) and a class 3B with centraliser 3 1+2. + 2.Alt(4). Finally, there are 21-elements in G whose cube is in the class 7C. Recall that Semisimple classes contains a unique class of elements of order 3, and these are self-centralising. Further L 2 (8) also contains a unique class of elements of order 3, whose centraliser is a cyclic subgroup of order 9. Hence there are three G-conjugacy classes of elements of order 21 whose cube is in 3C, a class with centraliser 21 × L 2 (8), a class with centraliser 63 × L 3 (2) and a class with centraliser 7 × 3 × 9. Semisimple classes Lemma 3.15. There are ten classes of elements of order 63 in G, with centralisers as in Table 2 . Semisimple classes Proof. Any element of order 63 has ninth power in either 7A, 7B or 7C in G. We begin by looking at the 7A case. Now C G (7A) ∼ = 7×L 6 (2), and there is a unique class of elements of order 9 in L 6 (2). These have centraliser of order 3 2 .7 in L 6 (2), and so we obtain a class of elements of order 63 with centraliser of shape 63 × 7. In the 7B case we have (2) contains no elements of order 9 but does have a unique class of elements of order 3, which are self-centralising. The group L 2 (8) has three self-centralising classes of elements of order 9, namely 9A, 9B 2 and 9C 4. Hence we obtain three classes of elements of order 63 with centraliser 63 × L 3 (2) and three class of elements of order 63 with centraliser 63 × 3. Semisimple classes Lemma 3.16. There are six classes of elements of order 217 in G. Each class is selfcentralising. Semisimple classes Proof. Examining the orders of the centralisers of elements of order 7 in G we see that only 7A-elements are centralised by elements of order 31. Furthermore, L 6 (2) has six classes of elements of order 31 (a master class and its slaves), all of which are self-centralising. Thus we see immediately that there are six classes of elements of order 217 in G, and these are all self-centralising. Semisimple classes Lemma 3.17. There is a unique class of elements of order 13 in G, and such elements have centraliser 13 × L 2 (8). Semisimple classes Proof. Note that E 6 (2) contains a full Sylow 13-subgroup of G. From [24] we know there is a single conjugacy class of elements of order 13 in E 6 (2), and hence a single class 13A of elements of order 13 in G. Moreover, we can deduce from [18] that these have centraliser 13 × L 2 (8), as claimed. Proof. Any element of order 117 in G powers down to a 13A-element. Again, using Lemma 3.17 and the fact that L 2 (8) contains three classes of elements of order 9 allows us to deduce that G contains three classes of elements of order 117 in G, all of which are self-centralising. Proof. Using [14] we see that the subgroup E 6 (2) contains a full Sylow 17-subgroup of G, and contains two classes of elements of order 17; in fact there is a unique class of cyclic subgroups of order 17 in E 6 (2). Hence there are at most two classes of elements of order 17 in G and the only possibility is that these two classes fuse in G. However, computation of Brauer character values on V yields 17-elements g 1 and g 2 with χ(g 1 ) = χ(g 2 ), where χ is the character afforded by the G-module V . Indeed, we find elements g 1 and g 2 of order 17 in G with Brauer character values Semisimple classes where ξ is a primitive seventeenth root of unity. Hence the potential fusion does not occur. Finally, we see from [18] that any element of order 17 in G has centraliser 17 × Alt(5) × Sym(3). Semisimple classes Lemma 3.22. There are six classes of elements of order 51 in G. Semisimple classes Proof. We know that Alt(5) contains a single conjugacy class of elements of order 3, as does Sym(3). Furthermore, in both these groups the 3-elements are self-centralising. Thus there are three classes of elements of order 51 which cube to 17A-elements. These have centralisers 51 × Alt(5), 51 × Sym(3) and 51 × 3. Likewise, we get three classes for elements of order 51 which cube to 17B-elements, accounting for the six classes claimed in the statement of the lemma. Semisimple classes Lemma 3.23. There is a unique class of elements of order 11 in G, and such elements have centraliser 11 × 3 1+2 : 2.Alt(4). Semisimple classes Proof. Using the Dynkin diagram of type E 7 , we see that G contains a subgroup Sp 10 (2). Such a subgroup contains a full Sylow 11-subgroup of G, and contains a unique conjugacy class of elements of order 11. Thus there is a unique class 11A in G. From Lemma 3.5 we have C G (3A) ∼ 3 Semisimple classes .2 E 6 (2).3. Further, this lemma implies that 3A-elements are conjugate in G to their inverses, and so we have N G (3A) ∼ 3 Semisimple classes .2 E 6 (2).Sym (3) . From the character table of 2 E 6 (2).Sym(3) (which is stored in [20] for example) we deduce that N G (3A) contains a unique class of elements of order 11. Now using Lemma 3.5 again we see that Semisimple classes Moreover, using the character table library in [20] we see that for any 11-element Semisimple classes .11, and we know from [18] that this is the order of the full centraliser in G. Proof. From Lemma 3.23 we know that any element of order 33 in G must cube to an 11A-element. Recall that C G (11A) ∼ 11 × 3 1+2 : 2.Alt(4). Now let g be an 11A-element in G and set K = C G (g). Let C denote the direct factor 3 1+2 : 2.Alt(4) of K, with C the factor group C/Z(C). Now C ∼ 3 2 : 2.Alt(4), and C has five classes of elements of order 3. One of these lies in the core of C and has centraliser of order 3 Semisimple classes 3 . This pulls back to C to yield a single class of elements of order 33 in G with centraliser of order 3 3 .11. There are two classes which have centraliser of size 2.3 2 in C. Each one of these classes pulls back to give three classes of elements of order 33 in G with centraliser of size 2.3 3 .11 (that is, there are six such classes in total). Finally, we need to consider pulling back the identity element of C. This yields two classes of elements of order 33 in G, both with centraliser 11 × 3 1+2 : 2.Alt(4). This accounts for nine classes of elements of order 33 in G. We now look at the elements of order 99 in G. Now there are two classes of elements of order 3 in C with centraliser of size 3 2 . These both pull back to give two classes of elements of order 9 and hence two G-classes of elements of order 99 in G with centraliser of size 3 2 .11. Semisimple classes Lemma 3.25. There are three classes of elements of order 43 in G. These have centraliser 43 × 3. Semisimple classes Proof. We know from Lemma 3.5 that for 3C-elements we have Semisimple classes Now U 7 (2) contains a full Sylow 43-subgroup of G, and calculations in U 7 (2) : 2 reveal that for any element g of order 43 the normaliser N U7(2):2 ( g ) is a Frobenius group 43 : 14. Hence we know that there are at most three classes of elements of order 43 in G. That these classes do not fuse in G is proved by the fact that we can compute three different Brauer character values for elements of order 43 in G on V , in a similar manner to the proof of Lemma 3.21. Finally, elements in each of these classes are clearly centralised by a 3C-element and that this is the full centraliser is apparent from [18] . Proof. This follows immediately from Lemma 3.25. Semisimple classes Lemma 3.27. There are nine classes of elements of order 127 in G and all are self-centralising. Semisimple classes Proof. We know from [27] that G contains (maximal) subgroups of the form L 2 (128) : 7. Now L 2 (128) : 7 contains elements of order 127, and if g is such an element then g must be a Sylow 127-subgroup of G. Thus any element of order 127 in G lies in a subgroup L 2 (128) : 7. Now any 127-element g ∈ L 2 (128) : 7 has normaliser 127 : 14 and hence contains 9 classes of elements of order 127. These do not fuse in G by Brauer character value considerations and account for all of the non-trivial semisimple classes of G whose centralisers are divisible by 127. That they are self-centralising again follows from [18] . Proof. From [14] we see that 19 divides | 2 E 6 (2)|, and hence Semisimple classes contains a full Sylow 19-subgroup of G. The character table of the group 2 E 6 (2).Sym(3) is in the Gap [20] character table library and we find that N G (3A) contains a unique class of elements of order 19. Hence we have a single class 19A of elements of order 19 in G. Now an element of order 19 in N G (3A) has centraliser 9 × 19 and using [18] we see that this is the order of the full centraliser in G. Proof. Any element of order 57 cubes to a 19A-element and C G (19A) ∼ = 19 × 9 from Lemma 3.28. Now a cyclic subgroup of order 9 contains two elements of order 3 which are non-conjugate since the group is abelian. Hence we obtain two classes of order 57 in G with centraliser as claimed. Semisimple classes Lemma 3.30. There are six classes of elements of order 171 in G and these are all selfcentralising. Semisimple classes Proof. We argue as in the proof of Lemma 3.29 but taking the six elements of order 9 in the cyclic subgroup. Proof. Note that Sp 10 (2) contains a full Sylow 31-subgroup of G. From the character table of Sp 10 (2) we know that it contains three classes 31ABC (master and slave classes). The possibility that these three classes fuse in E 7 (2) is negated by Brauer character computations, and the centraliser is found by referring to [18] . Proof. Any element of order 93 cubes into one of the three classes of elements of order 31 in G. Suppose this class is 31A. Then Semisimple classes , and since L 3 (2) has a unique class of elements of order 3 which are self-centralising, this yields a G-conjugacy class 93A which is self-centralising. Similarly, we get a class each for the cases 31B and 31C and so a total of three classes as claimed. Semisimple classes Lemma 3.33. There are four classes of elements of order 73 in G, all of which are selfcentralising. Semisimple classes Proof. Since 73 divides |E 6 (2)| we have that E 6 (2) contains a full Sylow 73-subgroup of G. From the character table of E 6 (2) we know it contains eight classes of elements of order 73 which are all self-centralising. Since for any E 6 (2)-subgroup of G the automorphism group E 6 (2) : 2 is also contained in G, these fuse to four classes in E 7 (2). Any further fusion is ruled out by the computation of Brauer characters. That these elements are self-centralising is seen by referring to [18] . Semisimple classes Lemma 3.34. There are six classes of elements of order 9 in G, with centralisers as in Table 2 . Moreover, elements from classes 9A and 9C cube into the class 3A, while elements from the remaining classes of elements of order 9 cube into the class 3E. Semisimple classes Proof. There are exactly 128 G-conjugacy classes of semisimple elements, and thus far we have accounted for 122 such classes. Therefore there are at most six classes of elements of order 9 in G. We attack this problem computationally, finding six representatives of elements of order 9 which are non-conjugate in G. We distinguish between elements in differing classes by comparing the dimensions of their fixed spaces on V or their Brauer character values. In each case we generate the centraliser by using the method of [7] . 4 . Almost simple subgroups in Lie (2) 4.1. Non-existence of certain subgroups Semisimple classes Proof. In each case we apply Lagrange's theorem to deduce that H cannot be contained in G. Semisimple classes . Then H contains an element of order 13, and such an element has cyclic centraliser of order 65 in H. However, by Lemma 3.17 there is a unique class of elements of order 13 in G and (8) , and is also contained in U 4 (8) (see [39] , for example), so these latter three groups also cannot be contained in G. Semisimple classes Proof. In each case we find that there is no possible restriction of Brauer characters of H to V , thus showing that H cannot be a subgroup of G. For the majority of cases this can be easily demonstrated using the Brauer character table of H, the relevant pieces of which are contained in Tables A.13, A.18, A.30 and A.32, and the values of the Brauer characters of G given in Table 3 . However if H ∼ = L 3 (16), then we note that H contains a maximal subgroup which is isomorphic to U 3 (4). It is now easily seen from the Brauer character table of U 3 (4) that such a subgroup cannot be contained in G, and so H G. Normalisers of cyclic subgroups of order 7 (Electronic files folder /Cyclic7Normalisers) In proving a number of results which follow we make use of normalisers of certain cyclic subgroups of order 7. Here we describe how to construct computationally the normalisers of subgroups X G, where X = x and x is an element from class 7B or 7C. First we have the case x ∈ 7C. We first construct the subgroup Normalisers of cyclic subgroups of order 7 (Electronic files folder /Cyclic7Normalisers) before finding x ∈ H such that x has order 7 and C V (x) has dimension 8 (such an x projects non-trivially onto both factors of H). Thus x ∈ 7C, and x is inverted by an involution t which lies in the Weyl group of G and swaps the root α 0 with α 1 and the root α 6 with α 7 . Set X = x , and note that [N G (X) : C G (X)] 6. To construct N G (X) computationally, we first construct the normaliser of X in a variety of subgroups generated by fundamental root generators, along with x ±α0 . Adding t to our generating set, this yields a subgroup K N G (X) where Normalisers of cyclic subgroups of order 7 (Electronic files folder /Cyclic7Normalisers) At this stage, to complete the construction of the full normaliser we might look to apply the method of [7] with the elements x and t. However, experimental evidence suggests that the method of [7] is unsuccessful when applied with these elements (or at least very inefficient). We therefore take an element y of order 7 in K such that C V (y) has dimension 20. Note that y may be chosen to also be inverted by t, so we may apply [7] with this pair of elements and construct Normalisers of cyclic subgroups of order 7 (Electronic files folder /Cyclic7Normalisers) We now check that Normalisers of cyclic subgroups of order 7 (Electronic files folder /Cyclic7Normalisers) which must be all of N G (X) by order considerations. For x ∈ 7B the method is very similar. In fact, we find that a direct application of [7] (using x and the inverting involution t) yields the full normaliser Normalisers of cyclic subgroups of order 7 (Electronic files folder /Cyclic7Normalisers) Lemma 4.4. Let X = x be a cyclic group of prime order, with W a finite dimensional KX-module, where K is some field. Suppose R, S are isomorphic irreducible X-submodules of W with dimension n, and that X acts transitively on the set of non-zero vectors of R, respectively S. Set u := r 0 +s 0 , where u = 0, r 0 ∈ R, s 0 ∈ S, and set U = u, ux, . . . , ux n−1 ⊆ W . Then U is an X-module, and is isomorphic to R (and S). Normalisers of cyclic subgroups of order 7 (Electronic files folder /Cyclic7Normalisers) Proof. When either r 0 = 0 or s 0 = 0 the result is clear, so assume that this is not the case. If R = S, then since X acts transitively on the non-zero vectors of R we have r 0 x i = u for some i, and multiplication by x i is an X-module isomorphism between R and U , as required. Suppose Normalisers of cyclic subgroups of order 7 (Electronic files folder /Cyclic7Normalisers) Since both R and S are irreducible we have R∩S = {0}, and so we must have r 0 x n−1 = n−2 i=0 k i r 0 x i , a contradiction since R is irreducible with dimension n. Thus dim(U ) = n. Let θ : R → S be an X-module isomorphism, which exists by assumption. Since X acts transitively on the non-zero vectors of S, we must have r 0 θx i = s 0 for a suitable choice of i, and since θx i : R → S is also an X-module isomorphism, without loss of generality we may assume that r 0 θ = s 0 . Normalisers of cyclic subgroups of order 7 (Electronic files folder /Cyclic7Normalisers) We now show that U is X-invariant. Clearly it suffices to show that ux n is a linear combination of {u, ux, . . . , ux n−1 }. Using the fact that θ is an X-module isomorphism, we have Normalisers of cyclic subgroups of order 7 (Electronic files folder /Cyclic7Normalisers) Now, since dim(R) = n we have Normalisers of cyclic subgroups of order 7 (Electronic files folder /Cyclic7Normalisers) Thus dim(U ) = n, as claimed. We now define a map φ : R → U by setting (r 0 x i )φ := ux i , for 0 i n − 1, and extending linearly. It is straightforward to check that φ is an X-module homomorphism, and the result follows. Normalisers of cyclic subgroups of order 7 (Electronic files folder /Cyclic7Normalisers) Before stating our next results we note that if X is a cyclic group of order 7 then there are exactly three isomorphism classes of irreducible X-modules over F 2 , namely one consisting of the trivial module, and two isomorphism classes of 3-dimensional modules. Normalisers of cyclic subgroups of order 7 (Electronic files folder /Cyclic7Normalisers) Lemma 4.5. Suppose X = x , where x ∈ 7C. As an X-module we have Normalisers of cyclic subgroups of order 7 (Electronic files folder /Cyclic7Normalisers) where Normalisers of cyclic subgroups of order 7 (Electronic files folder /Cyclic7Normalisers) Moreover, representatives for these orbits are included in the accompanying electronic file /RepsNx7C. Proof. It is straightforward to check that Stab , and this subgroup has index 2 in N G (X), with V t 1 = V 2 , where t is any involution which inverts x. Since |X| = 7 and any 3-dimensional subspace of V contains exactly 7 non-zero vectors, X must act transitively on the non-zero vectors of any 3-dimensional X-submodule of V . Therefore we may apply Lemma 4.4 to see that if 0 = v ∈ V 1 , then U = v, vx, vx 2 ∈ U 1 . Moreover, since U is irreducible it is the unique submodule in U 1 which contains v. Since there are 2 24 − 1 non-zero vectors in V 1 , we deduce that there are (2 24 − 1)/7 = 2 396 745 irreducible 3-dimensional X-submodules of V 1 (and similarly of V 2 ). Write S = Stab N G (X) (V 1 ). By making suitable choices of v and constructing the orbits of the corresponding U under the action of S (using Magma's Orbit command), we find a complete set of S-orbit representatives for U 1 , with lengths as given in the statement of the lemma. Note that since also S = Stab N G (X) (V 2 ), and V 2 is N G (X)-conjugate to V 1 , the S-orbits of U 1 will be in one-to-one correspondence with those of U 2 . Proof. It is straightforward to check that Stab Lemma 4.6. Suppose X = x , where x ∈ 7B. As an X-module we have Proof. It is straightforward to check that Stab https://www.cambridge.org/core/terms. https://doi.org/10.1112/S1461157015000030 Proof. It is straightforward to check that Stab where V 1 = 9 i=1 U i and V 2 = 9 j=1Ũ j . The subspaces U i lie in one isomorphism class of irreducible 3-dimensional X-modules, and the subspacesŨ j lie in the other. Denote by U k the set of irreducible 3-dimensional X-submodules of V which are contained in V k , for k = 1, 2. Proof. It is straightforward to check that
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