Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13096 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#13101 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1898 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C2×#41 | C23×C19 |
| prime radical | C2×C19×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C2×C4×C19 ◁ C2×C19×2−1+4 ◁ G | |
630 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)(20,21,26,32,27,33,25,31)(22,34,39,64,40,65,38,63)(23,41,44,29,45,57,43,58)(24,46,35,61,49,30,50,52)(28,51,54,47,55,73,60,37)(36,67,70,62,71,82,69,79)(42,76,66,56,48,75,77,59)(53,72,80,83,81,68,74,78);(20,22,35,51)(21,28,52,64)(23,36,66,78)(24,47,26,38)(25,39,49,37)(27,40,50,73)(29,53,75,67)(30,34,32,54)(31,60,46,65)(33,55,61,63)(41,74,59,62)(42,72,44,69)(43,70,48,68)(45,71,77,83)(56,79,57,80)(58,81,76,82);(20,23,27,45)(21,29,33,58)(22,36,40,71)(24,48,49,42)(25,44,26,43)(28,53,55,81)(30,59,46,56)(31,57,32,41)(34,62,65,79)(35,66,50,77)(37,72,47,68)(38,70,39,69)(51,78,73,83)(52,75,61,76)(54,74,60,80)(63,82,64,67)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6089);