Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13229 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#13233 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1964 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C2×#41 | C23×C19 |
| prime radical | C2×#41×C19 | C23 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C2×C4×C19 ◁ C2×C19×#131 ◁ G | |
432 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)(20,21,26,31,27,32,25,47)(22,33,37,60,38,61,48,59)(23,39,42,56,43,70,41,29)(24,44,34,57,46,30,49,51)(28,50,53,45,54,67,77,36)(35,63,65,58,66,80,74,81)(40,71,62,76,73,69,75,55)(52,64,68,78,79,72,83,82);(20,22,34,50)(21,28,51,60)(23,35,62,78)(24,45,26,48)(25,37,46,36)(27,38,49,67)(29,52,71,80)(30,33,31,53)(32,54,57,59)(39,68,55,58)(40,72,42,74)(41,65,73,64)(43,66,75,82)(44,61,47,77)(56,79,69,63)(70,83,76,81);(20,23,26,42,27,43,25,41)(21,29,47,70,32,56,31,39)(22,35,48,74,38,66,37,65)(24,40,49,75,46,73,34,62)(28,52,77,83,54,79,53,68)(30,55,51,71,44,76,57,69)(33,58,60,80,61,81,59,63)(36,64,50,78,45,72,67,82)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6221);