Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#11643 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C2×C22⋊C4 |
| commutator | C2×C4 | C22×C8×C19 |
| Frattini, cosocle | C4×C8 | C23×C19 |
| socle | C22×C19 | C2×C22⋊C8 |
| prime radical | C23×C19 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C4×C8×C19 ◁ G | |
288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 3 generators on 67 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,29)(23,27)(24,33)(25,28)(26,31)(30,34)(32,35)(36,37,40,46,41,47,55,64,42,44,56,60,39,45,53,62)(38,48,50,65,49,58,66,63,51,61,54,67,52,43,57,59);(20,22,25,32)(21,26,28,34)(23,31,24,30)(27,35,33,29)(36,38)(37,43)(39,52)(40,54)(41,49)(42,51)(44,58)(45,61)(46,63)(47,48)(50,56)(53,66)(55,57)(59,60)(62,65)(64,67);(20,23,25,24)(21,27,28,33)(22,30,32,31)(26,29,34,35)(37,44)(43,58)(45,47)(46,60)(48,61)(59,63)(62,64)(65,67)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4631);