Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#13636 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | C2×D16 |
| commutator | C16 | C22×C4×C19 |
| Frattini, cosocle | C2×C16 | C23×C19 |
| socle | C2×C19 | C2×#366 |
| prime radical | C23×C19 | Q32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C22×C8×C19 ◁ C22×C16×C19 ◁ G | |
178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 83 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,24,30,27,32,26,48)(22,33,37,59,35,57,39,70)(23,40,42,71,44,73,43,74)(25,46,45,56,47,75,41,31)(28,49,51,64,50,67,76,36)(29,53,54,63,55,62,72,34)(38,69,68,80,65,83,66,60)(52,61,58,77,78,81,79,82);(20,22,34,61,25,38,42,49,24,37,53,77,45,68,44,64,27,35,63,81,47,65,43,67,26,39,62,82,41,66,23,36)(21,28,40,60,31,52,72,70,48,76,74,83,75,79,55,57,32,50,73,80,56,78,54,59,30,51,71,69,46,58,29,33);(22,35)(28,50)(33,57)(36,64)(37,39)(38,65)(49,67)(51,76)(52,78)(58,79)(59,70)(60,80)(61,81)(66,68)(69,83)(77,82)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6624);