Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#18972 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C19 |
| prime radical | C19×#121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C2×C42×C19 ◁ G | |
818 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)(23,30)(26,32)(27,31)(29,34)(33,35)(36,37,42,47)(38,43,50,58)(39,51,53,61)(40,46,41,45)(44,55,59,52)(48,62,64,66)(49,60,54,57)(56,63,67,65);(20,21,24,27)(22,26,28,33)(23,31,30,25)(29,35,34,32)(36,38)(37,43)(39,48)(40,49)(41,54)(42,50)(44,56)(45,57)(46,60)(47,58)(51,66)(52,63)(53,64)(55,65)(59,67)(61,62);(20,22,24,28)(21,26,27,33)(23,29,30,34)(25,32,31,35)(36,39)(37,44)(38,48)(40,52)(41,55)(42,53)(43,56)(45,51)(46,61)(47,59)(49,63)(50,64)(54,65)(57,66)(58,67)(60,62);(22,28)(26,33)(29,34)(32,35)(36,40,42,41)(37,45,47,46)(38,49,50,54)(39,52,53,55)(43,57,58,60)(44,51,59,61)(48,63,64,65)(56,66,67,62)703 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 11960);