group has order 17971200 = 211 ⋅ 33 ⋅ 52 ⋅ 13. Schur multiplier: Trivial for n ≥ 1 and for the Tits group. Outer automorphism group: 1⋅f⋅1, where f = 2n + 1. Order 2 for the Tits group. Remarks: Unlike the other simple groups of Lie type, the Tits group does not have a BN pair, though its automorphism group does so most authors count it as a sort of honorary group of Lie type. Ree groups, 2G2(32n+1) Simplicity: Simple for n ≥ 1. The group 2G2(3) is not simple, but its derived group 2G2(3)′ is a simple subgroup of index 3. Order: q3 (q3 + 1) (q − 1), where q = 32n+1 Schur multiplier: Trivial for n ≥ 1 and for 2G2(3)′. Outer automorphism group: 1⋅f⋅1, where f = 2n + 1. Other names: Ree(32n+1), R(32n+1), E2∗(32n+1) . Isomorphisms: The derived group 2G2(3)′ is isomorphic to A1(8). Remarks: 2G2(32n+1) has a doubly transitive permutation representation on 33(2n+1) + 1 points and acts on a 7-dimensional vector space over the field with 32n+1 elements. Sporadic groups Mathieu groups, M11, M12, M22, M23, M24 Janko groups, J1, J2, J3, J4 Conway groups, Co1, Co2, Co3 Fischer groups, Fi22, Fi23, Fi24′ Higman–Sims group, HS Order: 29 ⋅ 32 ⋅ 53 ⋅ 7 ⋅ 11 = 44352000 Schur multiplier: Order 2. Outer automorphism group: Order 2. Remarks: It acts as a rank 3 permutation group on the Higman Sims graph with 100 points, and is contained in Co2 and in Co3. McLaughlin group, McL Order: 27 ⋅ 36 ⋅ 53 ⋅ 7 ⋅ 11 = 898128000 Schur multiplier: Order 3. Outer automorphism group: Order 2. Remarks: Acts as a rank 3 permutation group on the McLaughlin graph with 275 points, and is contained in Co2 and in Co3. Held group, He Order: 210 ⋅ 33 ⋅ 52 ⋅ 73 ⋅ 17 = 4030387200 Schur multiplier: Trivial. Outer automorphism group: Order 2. Other names: Held–Higman–McKay group, HHM, F7, HTH Remarks: Centralizes an element of order 7 in the monster group. Rudvalis group, Ru Order: 214 ⋅ 33 ⋅ 53 ⋅ 7 ⋅ 13 ⋅ 29 = 145926144000 Schur multiplier: Order 2. Outer automorphism group: Trivial. Remarks: The double cover acts on a 28-dimensional lattice over the Gaussian integers. Suzuki sporadic group, Suz Order: 213 ⋅ 37 ⋅ 52 ⋅ 7 ⋅ 11 ⋅ 13 = 448345497600 Schur multiplier: Order 6. Outer