Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#13146 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #466 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1934 |
| prime radical | C2×C19×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C19×#405 ◁ G | |
312 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)(20,21,25,31,27,32,26,49)(22,33,36,61,38,62,37,71)(23,39,42,74,44,56,43,76)(24,45,46,51,48,77,47,30)(28,50,53,70,54,69,60,35)(29,55,57,40,58,75,73,41)(34,63,66,79,68,81,67,52)(59,80,82,64,72,78,83,65);(20,22,24,35,25,36,46,50,27,38,48,70,26,37,47,69)(21,28,51,62,49,60,45,61,32,54,30,33,31,53,77,71)(23,34,40,64,42,66,75,78,44,68,41,65,43,67,55,80)(29,52,39,72,73,81,76,82,58,79,56,59,57,63,74,83);(20,23,24,40,25,42,46,75,27,44,48,41,26,43,47,55)(21,29,30,56,49,73,77,74,32,58,51,39,31,57,45,76)(22,34,35,64,36,66,50,78,38,68,70,65,37,67,69,80)(28,52,33,59,60,81,71,83,54,79,62,72,53,63,61,82)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6134);