Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#11158 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #397 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| prime radical | C24×C19 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C23×C4×C19 ◁ G | |
794 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)(20,21)(22,23)(24,25)(26,35)(27,40,44,55)(28,45)(29,33)(30,34)(31,37)(32,43,48,42)(36,51,41,46)(38,47)(39,49)(50,53,54,52);(24,26,30,39)(25,31,34,47)(27,41,44,36)(28,38,29,37)(32,46,48,51)(33,49,45,35)(40,54,55,50)(42,52,43,53);(21,23)(24,27,28,42,30,44,29,43)(25,32,33,40,34,48,45,55)(26,36,37,52,39,41,38,53)(31,46,35,50,47,51,49,54)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4146);