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 | #72268062 | A5 |
| LCS | G | |
| UCS | C1 ◁ C22 | |
12547 conjugacy classes of subgroups; only the 11 normal subgroups are shown.
This group has 12547 conjugacy classes of subgroups — too many to draw. Only its 11 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) = 768 |
A minimal faithful permutation representation — 2 generators on 112 points:
(1,4,14,7)(2,8,5,16)(3,11,26,23)(6,19,9,12)(10,25,21,15)(13,22,32,30)(17,31,27,20)(18,24,29,28)(34,37,46,64)(35,40,38,48)(36,43,59,83)(39,50,71,82)(42,56,85,108)(45,58,87,107)(47,67,96,75)(49,69,61,68)(51,74,72,99)(52,76,97,105)(53,79,95,63)(54,81,65,94)(55,84,78,104)(57,60,77,102)(62,91,89,70)(66,92,86,88)(73,100,98,80)(90,101,112,111);(1,21,19,25,22,8)(2,13,30,15,9,11)(3,24,10,14,27,31)(4,12)(5,23,18,26,20,7)(6,17)(16,29)(28,32)(33,100,42,88,76,84)(34,103,54,102,71,96)(35,43,66,80,60,44)(36,79,78,37,77,89)(38,91,93,112,45,92)(39,99,83,109,61,95)(40,69,67,75,111,97)(41,105,46,86,90,53)(47,70,50,49,85,110)(48,52,94,72,51,98)(55,56,64)(58,59,62)(65,87,106,74,104,82)(68,73)(81,101)(107,108)63 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((1,4,14,7)(2,8,5,16)(3,11,26,23)(6,19,9,12)(10,25,21,15)(13,22,32,30)(17,31,27,20)(18,24,29,28)(34,37,46,64)(35,40,38,48)(36,43,59,83)(39,50,71,82)(42,56,85,108)(45,58,87,107)(47,67,96,75)(49,69,61,68)(51,74,72,99)(52,76,97,105)(53,79,95,63)(54,81,65,94)(55,84,78,104)(57,60,77,102)(62,91,89,70)(66,92,86,88)(73,100,98,80)(90,101,112,111), (1,21,19,25,22,8)(2,13,30,15,9,11)(3,24,10,14,27,31)(4,12)(5,23,18,26,20,7)(6,17)(16,29)(28,32)(33,100,42,88,76,84)(34,103,54,102,71,96)(35,43,66,80,60,44)(36,79,78,37,77,89)(38,91,93,112,45,92)(39,99,83,109,61,95)(40,69,67,75,111,97)(41,105,46,86,90,53)(47,70,50,49,85,110)(48,52,94,72,51,98)(55,56,64)(58,59,62)(65,87,106,74,104,82)(68,73)(81,101)(107,108));