Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#19531 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C2×#121 |
| commutator, Frattini, cosocle | C42 | C24×C19 |
| prime radical | C23×C19 | #121 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C2×C42×C19 ◁ G | |
536 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 4 generators on 67 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,25)(22,29)(23,26)(28,33)(30,31)(32,34)(36,37,41,45)(38,46,50,60)(39,43,51,57)(40,53,47,44)(42,54,56,49)(48,59,62,67)(52,64,61,58)(55,65,66,63);(20,21,26,33,24,28,23,25)(22,27,29,34,30,35,31,32)(36,38,47,54,41,50,40,49)(37,42,53,60,45,56,44,46)(39,48,61,65,51,62,52,63)(43,55,64,67,57,66,58,59);(20,22,24,30)(21,27,28,35)(23,31,26,29)(25,32,33,34)(36,39)(37,43)(38,48)(40,52)(41,51)(42,55)(44,58)(45,57)(46,59)(47,61)(49,63)(50,62)(53,64)(54,65)(56,66)(60,67);(21,28)(22,30)(25,33)(29,31)(39,51)(43,57)(48,62)(52,61)(55,66)(58,64)(59,67)(63,65)703 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 12519);