Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 5 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#13620 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | C2×D16 |
| commutator | C16 | C22×C4×C19 |
| Frattini, cosocle | C2×C16 | C23×C19 |
| socle | C2×C19 | C2×#357 |
| prime radical | C2×C19×#508 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C22×C8×C19 ◁ C22×C16×C19 ◁ G | |
290 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)(20,21,24,29,26,32,45,31)(22,33,36,63,34,62,38,69)(23,39,42,70,44,72,77,71)(25,46,41,30,48,58,76,60)(27,49,51,64,50,66,61,35)(28,53,55,78,57,80,56,43)(37,68,67,81,65,82,47,54)(40,59,74,79,73,83,75,52);(20,22)(21,27)(23,40)(24,36)(25,47)(26,34)(28,54)(29,51)(30,59)(31,61)(32,50)(33,39)(35,43)(37,41)(38,45)(42,74)(44,73)(46,52)(48,67)(49,53)(55,68)(56,82)(57,81)(58,79)(60,83)(62,72)(63,70)(64,78)(65,76)(66,80)(69,71)(75,77);(22,34)(27,50)(33,62)(35,64)(36,38)(37,65)(40,73)(47,67)(49,66)(51,61)(52,79)(54,81)(59,83)(63,69)(68,82)(74,75)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6608);