Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#12586 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C2×#99 |
| commutator | C2×C4 | C2×C42×C19 |
| Frattini, cosocle | C22×Q8 | C23×C19 |
| prime radical | C24×C19 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C24×C19 ◁ C23×Q8×C19 ◁ G | |
746 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)(24,25,28,32,29,33,43,49)(26,34,37,51,35,50,41,54)(27,38,39,48,40,31,42,47)(30,44,46,52,45,53,55,36);(20,21,22,23)(24,26)(25,30,33,45)(27,36)(28,41)(29,35)(31,34,38,50)(32,46,49,55)(37,43)(39,53)(40,52)(42,44)(47,51,48,54);(26,35)(30,45)(34,50)(36,52)(37,41)(44,53)(46,55)(51,54)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5574);