Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13156 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#13157 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1936 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C23×C19 | #137 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C22×C4×C19 ◁ C19×#405 ◁ G | |
296 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)(20,21,26,33,28,34,27,56)(22,35,39,58,40,67,54,68)(23,41,45,63,46,69,55,77)(24,47,25,51,50,32,49,31)(29,38,59,37,61,52,60,48)(30,43,64,44,66,78,65,53)(36,71,74,82,75,57,42,62)(70,79,80,73,83,72,81,76);(20,22,24,48,27,39,49,38,28,40,50,37,26,54,25,52)(21,29,51,35,33,60,32,68,34,61,31,67,56,59,47,58)(23,42,78,73,45,75,53,79,46,74,43,76,55,36,44,72)(30,62,63,81,65,71,41,70,66,82,77,80,64,57,69,83);(20,23,24,43,27,45,49,44,28,46,50,78,26,55,25,53)(21,30,31,63,33,65,47,41,34,66,51,77,56,64,32,69)(22,36,37,72,39,42,52,73,40,75,48,79,54,74,38,76)(29,57,35,70,60,62,68,80,61,71,67,83,59,82,58,81)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6145);