Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#13126 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C19 | #124 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1924 |
| prime radical | C2×#41×C19 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C19 ◁ C22×C8×C19 ◁ G | |
300 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)(20,21,25,31,26,32,49,64,27,33,50,65,24,30,35,52)(22,34,39,66,37,56,67,62,40,54,47,78,38,28,51,55)(23,41,45,74,44,61,68,63,46,59,48,79,43,29,57,60)(36,69,70,81,42,75,76,83,71,53,73,80,72,58,77,82);(20,22,35,67,24,38,50,39,27,40,49,51,26,37,25,47)(21,28,52,66,30,54,65,55,33,56,64,78,32,34,31,62)(23,42,57,73,43,36,48,76,46,72,68,70,44,71,45,77)(29,58,79,81,59,53,63,82,61,75,74,80,41,69,60,83);(20,23,24,43,27,46,26,44)(21,29,30,59,33,61,32,41)(22,36,37,42,40,71,38,72)(25,48,35,68,50,45,49,57)(28,53,34,58,56,69,54,75)(31,63,52,74,65,60,64,79)(39,73,67,77,47,70,51,76)(55,81,66,83,62,80,78,82)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6114);