Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#19013 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C19 |
| prime radical | C19×#3531 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C2×C42×C19 ◁ G | |
810 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)(20,21)(22,29)(23,27)(24,32)(25,28)(26,30)(31,34)(33,35)(36,37,42,45)(38,48,51,60)(39,52,54,61)(40,47,41,46)(43,55,58,50)(44,56,59,53)(49,57,65,67)(62,63,66,64);(20,21)(22,26)(23,27)(24,32)(25,28)(29,33)(30,34)(31,35)(36,38,42,51)(37,43,45,58)(39,49,54,65)(40,55,41,50)(44,57,59,67)(46,60,47,48)(52,66,61,62)(53,64,56,63);(20,22)(21,26)(23,30)(24,33)(25,31)(27,34)(28,35)(29,32)(36,39)(37,44)(38,49)(40,53)(41,56)(42,54)(43,57)(45,59)(46,52)(47,61)(48,62)(50,63)(51,65)(55,64)(58,67)(60,66);(20,23,25,24)(21,27,28,32)(22,30,31,33)(26,34,35,29)(37,45)(43,58)(44,59)(46,47)(48,60)(52,61)(57,67)(62,66)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 12001);