Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#11188 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #394 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| prime radical | C24×C19 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C23×C4×C19 ◁ G | |
794 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,23)(24,25,29,34,30,33,28,45)(26,35,38,51,39,31,37,47)(27,40,43,49,44,50,42,32)(36,48,41,46,54,55,53,52);(24,26,30,39)(25,31,33,35)(27,41,44,53)(28,38,29,37)(32,48,49,55)(34,51,45,47)(36,42,54,43)(40,52,50,46);(21,23)(24,27)(25,32,33,49)(26,36)(28,42)(29,43)(30,44)(31,46,35,52)(34,40,45,50)(37,53)(38,41)(39,54)(47,55,51,48)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4176);