Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#11288 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #120 |
| commutator | C24 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C22×C19 | #385 |
| prime radical | C24×C19 | C22×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ G | |
602 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)(24,25,28,33,29,32,41,47)(26,30,36,44,35,43,39,48)(27,31,38,46,37,45,40,49)(34,42,51,53,50,54,52,55);(20,21)(22,23)(24,26)(25,30,32,43)(27,34,37,50)(28,39)(29,35)(31,42)(33,48,47,44)(36,41)(38,52,40,51)(45,54)(46,55)(49,53);(21,23)(24,27)(25,31,32,45)(26,34,35,50)(28,40)(29,37)(30,42)(33,49,47,46)(36,52,39,51)(38,41)(43,54)(44,55)(48,53)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4276);