Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#13243 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #466 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1970 |
| prime radical | C2×C19×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C19×#405 ◁ G | |
410 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)(22,34,39,58)(23,41)(24,31)(25,48)(26,50)(27,33)(28,37,53,46)(29,43)(30,35)(32,49)(36,51,66,77)(38,54,62,76)(40,52,68,61)(42,70)(44,55)(45,72)(47,59)(56,74)(57,63)(60,80,79,65)(64,81,73,69)(67,83,82,78)(71,75);(20,22,35,62,49,68,24,46,27,39,25,38,26,40,47,37)(21,28,31,52,50,76,30,58,33,53,59,61,32,54,48,34)(23,36,63,80,75,82,42,73,45,66,43,65,44,67,74,64)(29,51,41,69,70,83,55,79,57,77,72,81,56,78,71,60);(20,23,24,42,26,44,35,63,27,45,47,74,49,75,25,43)(21,29,48,71,32,56,59,72,33,57,30,55,50,70,31,41)(22,36,46,73,40,67,62,80,39,66,37,64,68,82,38,65)(28,51,34,60,54,78,61,81,53,77,58,79,76,83,52,69)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6231);