Finite Group DatabasebetaFinite Group Databasebeta| order | 61440 |
| automorphism | ? |
| homology | Abelianization (H1) = C1 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, hypercenter | C22 | #72209721 |
| commutator, hypocenter | G | C1 |
| Fitting, cosocle | 2−1+42 | A5 |
| LCS | G | |
| UCS | C1 ◁ C22 | |
At least 8357 conjugacy classes of subgroups; only the 12 normal subgroups are shown.
This group has at least 8357 conjugacy classes of subgroups — too many to draw. Only its 12 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 112 |
| transitive degree | μt(G) = 1280 |
A minimal faithful permutation representation — 2 generators on 112 points:
(1,3,10,15)(2,6,16,8)(4,9,12,22)(5,14,13,20)(7,19,30,29)(11,17,27,26)(18,25,32,28)(21,23,24,31)(33,35)(34,38,40,44)(36,43,48,64)(37,45)(39,50,49,65)(41,53,58,79)(42,55,73,98)(46,59,81,74)(47,57)(51,67,62,83)(52,68,63,85)(54,70,84,93)(56,76,101,108)(60,61,71,78)(66,89,104,103)(69,94,86,88)(72,97,105,112)(75,100,106,80)(77,99,90,95)(82,96)(87,102,91,109)(92,110,107,111);(1,3,8,2)(4,27,9,13,24,12,20,30,5,22,32,14)(6,25,31,15,29,17,10,23,28,16,26,19)(7,21,11,18)(33,89,64,60,77,98)(34,102,70,45,90,67)(35,109,79,71,75,85)(36,42,54,84,49,58)(37,106,50,40,103,59)(38,112,65,82,88,93)(39,63)(41,52,46,81,62,48)(43,47,110,68,78,94)(44,111,83,96,108,74)(51,73)(53,57,97,55,61,76)(56,104,107)(66,101,100,91,86,99)(69,87,105)(72,92)68 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,3,10,15)(2,6,16,8)(4,9,12,22)(5,14,13,20)(7,19,30,29)(11,17,27,26)(18,25,32,28)(21,23,24,31)(33,35)(34,38,40,44)(36,43,48,64)(37,45)(39,50,49,65)(41,53,58,79)(42,55,73,98)(46,59,81,74)(47,57)(51,67,62,83)(52,68,63,85)(54,70,84,93)(56,76,101,108)(60,61,71,78)(66,89,104,103)(69,94,86,88)(72,97,105,112)(75,100,106,80)(77,99,90,95)(82,96)(87,102,91,109)(92,110,107,111), (1,3,8,2)(4,27,9,13,24,12,20,30,5,22,32,14)(6,25,31,15,29,17,10,23,28,16,26,19)(7,21,11,18)(33,89,64,60,77,98)(34,102,70,45,90,67)(35,109,79,71,75,85)(36,42,54,84,49,58)(37,106,50,40,103,59)(38,112,65,82,88,93)(39,63)(41,52,46,81,62,48)(43,47,110,68,78,94)(44,111,83,96,108,74)(51,73)(53,57,97,55,61,76)(56,104,107)(66,101,100,91,86,99)(69,87,105)(72,92));