Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#12824 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #476 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C2×C19×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ C2×C4×C8×C19 ◁ G | |
414 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)(21,23)(24,25)(26,36)(27,40,42,47)(28,44)(29,31)(30,35)(32,38,50,54)(33,37)(34,39)(41,53,43,52)(45,49,55,51)(46,48);(20,21,22,23)(24,26,37,46,30,39,28,29)(25,31,44,34,35,48,33,36)(27,38,43,55,42,54,41,45)(32,47,51,52,50,40,49,53);(24,27)(25,32)(26,38)(28,41)(29,45)(30,42)(31,47)(33,49)(34,52)(35,50)(36,53)(37,43)(39,54)(40,48)(44,51)(46,55)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5812);