Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1426 |
| automorphism | ? |
| homology | Abelianization (H1) = C42×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#7476 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #367 |
| commutator | C16 | C42×C19 |
| Frattini, cosocle | C22×C16 | C22×C19 |
| socle | C2×C19 | #1380 |
| prime radical | C23×C19 | Q32 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C22×C8×C19 ◁ C22×C16×C19 ◁ G | |
162 conjugacy classes of subgroups.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 2 generators on 83 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,24,31,28,34,27,53)(22,35,37,67,40,69,46,68)(23,32,42,62,45,65,44,79)(25,48,47,64,43,30,50,61)(26,51,41,63,49,60,52,33)(29,54,55,36,59,71,58,73)(38,76,74,57,72,66,80,82)(39,78,75,56,77,81,70,83);(20,22,25,38,49,77,42,54,24,46,47,80,52,75,45,73,28,40,43,72,26,39,44,71,27,37,50,74,41,70,23,36)(21,29,32,56,63,82,61,68,53,55,79,81,51,76,30,35,34,59,65,83,33,57,64,67,31,58,62,78,60,66,48,69)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 464);