Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#11343 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #120 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C22×C19 | #385 |
| prime radical | C24×C19 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C23×C4×C19 ◁ G | |
666 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)(24,25,29,33,30,34,28,43)(26,35,39,46,38,51,37,31)(27,32,41,49,42,48,44,47)(36,50,52,55,40,54,53,45);(24,26,28,37,30,38,29,39)(25,31,33,35,34,46,43,51)(27,40,41,53,42,36,44,52)(32,45,47,54,48,55,49,50);(20,21,22,23)(24,27)(25,32)(26,36)(28,44)(29,41)(30,42)(31,45)(33,49)(34,48)(35,50)(37,53)(38,40)(39,52)(43,47)(46,55)(51,54)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4331);