Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C29 · Schur multiplier (H2) = C234 |
| direct product | C22×#72232065 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C27 |
| commutator, Frattini, cosocle | C23 | C29 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
30221771 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 9 generators on 24 points:
(17,18,21,22)(19,20,23,24);(6,8)(18,22)(20,24);(9,10)(11,14)(12,15)(13,16)(17,19)(18,20)(21,23)(22,24);(6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,20)(21,23)(22,24);(3,4);(6,8)(17,19)(18,20)(21,23)(22,24);(1,2)(3,4);(9,11)(10,14)(12,16)(13,15);(5,6)(7,8)(9,10)(11,13)(12,15)(14,16)(17,20)(18,23)(19,22)(21,24)784 conjugacy classes. The character table is too large to display.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((17,18,21,22)(19,20,23,24), (6,8)(18,22)(20,24), (9,10)(11,14)(12,15)(13,16)(17,19)(18,20)(21,23)(22,24), (6,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,20)(21,23)(22,24), (3,4), (6,8)(17,19)(18,20)(21,23)(22,24), (1,2)(3,4), (9,11)(10,14)(12,16)(13,15), (5,6)(7,8)(9,10)(11,13)(12,15)(14,16)(17,20)(18,23)(19,22)(21,24));