Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C28 · Schur multiplier (H2) = C231 |
| direct product | C22×#72233780 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C26 |
| commutator, Frattini, cosocle | C24 | C28 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
127348715 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 8 generators on 24 points:
(10,12)(14,16)(17,18)(19,22)(20,23)(21,24);(17,19)(18,21)(20,24)(22,23);(6,8)(10,12)(14,16)(17,19,20,24)(18,22,23,21);(6,8)(10,12)(17,19)(18,22)(20,24)(21,23);(3,4);(10,12);(1,2)(3,4);(5,6)(7,8)(9,10,11,12)(13,14)(15,16)(17,18)(19,21)(20,23)(22,24)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((10,12)(14,16)(17,18)(19,22)(20,23)(21,24), (17,19)(18,21)(20,24)(22,23), (6,8)(10,12)(14,16)(17,19,20,24)(18,22,23,21), (6,8)(10,12)(17,19)(18,22)(20,24)(21,23), (3,4), (10,12), (1,2)(3,4), (5,6)(7,8)(9,10,11,12)(13,14)(15,16)(17,18)(19,21)(20,23)(22,24));