automorphism group: Order 2. Other names: Sz Remarks: The 6 fold cover acts on a 12-dimensional lattice over the Eisenstein integers. It is not related to the Suzuki groups of Lie type. O'Nan group, O'N Order: 29 ⋅ 34 ⋅ 5 ⋅ 73 ⋅ 11 ⋅ 19 ⋅ 31 = 460815505920 Schur multiplier: Order 3. Outer automorphism group: Order 2. Other names: O'Nan–Sims group, O'NS, O–S Remarks: The triple cover has two 45-dimensional representations over the field with 7 elements, exchanged by an outer automorphism. Harada–Norton group, HN Order: 214 ⋅ 36 ⋅ 56 ⋅ 7 ⋅ 11 ⋅ 19 = 273030912000000 Schur multiplier: Trivial. Outer automorphism group: Order 2. Other names: F5, D Remarks: Centralizes an element of order 5 in the monster group. Lyons group, Ly Order: 28 ⋅ 37 ⋅ 56 ⋅ 7 ⋅ 11 ⋅ 31 ⋅ 37 ⋅ 67 = 51765179004000000 Schur multiplier: Trivial. Outer automorphism group: Trivial. Other names: Lyons–Sims group, LyS Remarks: Has a 111-dimensional representation over the field with 5 elements. Thompson group, Th Order: 215 ⋅ 310 ⋅ 53 ⋅ 72 ⋅ 13 ⋅ 19 ⋅ 31 = 90745943887872000 Schur multiplier: Trivial. Outer automorphism group: Trivial. Other names: F3, E Remarks: Centralizes an element of order 3 in the monster, and is contained in E8(3), so has a 248-dimensional representation over the field with 3 elements. Baby Monster group, B Order: 241 ⋅ 313 ⋅ 56 ⋅ 72 ⋅ 11 ⋅ 13 ⋅ 17 ⋅ 19 ⋅ 23 ⋅ 31 ⋅ 47 = 4154781481226426191177580544000000 Schur multiplier: Order 2. Outer automorphism group: Trivial. Other names: F2 Remarks: The double cover is contained in the monster group. It has a representation of dimension 4371 over the complex numbers (with no nontrivial invariant product), and a representation of dimension 4370 over the field with 2 elements preserving a commutative but non-associative product. Fischer–Griess Monster group, M Order: 246 ⋅ 320 ⋅ 59 ⋅ 76 ⋅ 112 ⋅ 133 ⋅ 17 ⋅ 19 ⋅ 23 ⋅ 29 ⋅ 31 ⋅ 41 ⋅ 47 ⋅ 59 ⋅ 71 = 808017424794512875886459904961710757005754368000000000 Schur multiplier: Trivial. Outer automorphism group: Trivial. Other names: F1, M1, Monster group, Friendly giant, Fischer's monster. Remarks: Contains all but 6 of the other sporadic groups as subquotients. Related to monstrous moonshine. The monster is the automorphism group of the 196,883-dimensional Griess algebra and the infinite-dimensional monster vertex operator algebra, and acts naturally on the