Finite Group DatabasebetaFinite Group Databasebeta| order | 4096 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C210 · Schur multiplier (H2) = C243 |
| direct product | C24×#62817 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C26 |
| commutator, Frattini, cosocle | C22 | C210 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
319431739 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 10 generators on 24 points:
(3,4);(5,6);(7,8);(9,10)(11,15)(12,13)(14,16)(17,18)(19,20)(21,22)(23,24);(17,19)(18,20)(21,24)(22,23);(10,13)(14,15)(18,22)(20,23);(1,2)(3,4)(5,6)(7,8);(18,22)(20,23);(9,11)(10,14)(12,16)(13,15)(17,19)(18,23)(20,22)(21,24);(9,10)(11,14)(12,13)(15,16)(17,20,21,23)(18,19,22,24)1216 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((3,4), (5,6), (7,8), (9,10)(11,15)(12,13)(14,16)(17,18)(19,20)(21,22)(23,24), (17,19)(18,20)(21,24)(22,23), (10,13)(14,15)(18,22)(20,23), (1,2)(3,4)(5,6)(7,8), (18,22)(20,23), (9,11)(10,14)(12,16)(13,15)(17,19)(18,23)(20,22)(21,24), (9,10)(11,14)(12,13)(15,16)(17,20,21,23)(18,19,22,24));