Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#11134 |
| 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 | |
838 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)(24,25,29,33,30,34,28,35)(26,36,40,48,39,47,38,31)(27,41,44,51,45,32,43,50)(37,49,53,46,52,54,42,55);(20,21)(22,23)(24,26,28,38,30,39,29,40)(25,31,33,36,34,48,35,47)(27,42,43,52,45,53,44,37)(32,49,50,46,41,54,51,55);(21,23)(24,27,30,45)(25,32,34,41)(26,37,39,52)(28,44,29,43)(31,46,48,55)(33,51,35,50)(36,49,47,54)(38,53,40,42)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4122);