Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#23368 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C19 | C24 |
| commutator | C23 | C23×C4×C19 |
| Frattini, cosocle | C22×C4 | C24×C19 |
| socle, prime radical | C23×C19 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C19 ◁ G | |
580 conjugacy classes of subgroups.
| minimal degree | μ(G) = 59 |
| transitive degree | μt(G) = 4864 |
A minimal faithful permutation representation — 4 generators on 59 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,26)(23,25)(24,27)(28,29)(30,37)(31,35)(32,41)(33,36)(34,40)(38,43)(39,42)(44,45,48,51,47,50,55,58)(46,52,54,59,53,49,56,57);(22,27)(24,26)(28,30,33,40)(29,34,36,37)(31,39,32,38)(35,43,41,42);(20,22,23,27)(21,24,25,26)(28,31,33,32)(29,35,36,41)(30,38,40,39)(34,42,37,43)(44,46,47,53)(45,49,50,52)(48,54,55,56)(51,57,58,59);(21,25)(24,26)(28,32,33,31)(29,35,36,41)(30,39,40,38)(34,42,37,43)(46,53)(49,52)(54,56)(57,59)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 16356);