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×#13096 |
| 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 | |
624 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)(22,32,37,61)(23,38,41,57)(24,42,33,29)(25,30)(26,31)(27,35)(28,40,53,47)(34,63)(36,60,46,77)(39,72)(43,51)(44,71)(45,74,62,55)(48,58)(49,78,81,83)(50,68)(52,73)(54,76)(56,75)(59,69)(64,70,67,80)(65,79)(66,82);(20,22,33,35,26,37,24,43)(21,27,42,61,31,51,29,32)(23,34,39,67,41,66,44,64)(25,46,62,75,48,36,45,68)(28,49,52,79,53,81,54,59)(30,56,74,60,58,50,55,77)(38,70,71,63,57,80,72,82)(40,69,73,83,47,65,76,78);(20,23)(21,28)(22,34)(24,44)(25,47)(26,41)(27,49)(29,54)(30,57)(31,53)(32,59)(33,39)(35,67)(36,69)(37,66)(38,58)(40,48)(42,52)(43,64)(45,73)(46,65)(50,70)(51,81)(55,71)(56,80)(60,82)(61,79)(62,76)(63,77)(68,83)(72,74)(75,78)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6084);