Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13417 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#13418 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #2041 |
| commutator, Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C2×C19×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C22×C4×C19 ◁ C19×#405 ◁ G | |
422 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)(22,36)(23,43)(24,34)(25,50)(26,32)(27,31)(28,35)(29,38)(30,46)(33,52)(37,69)(39,56)(40,57)(41,59)(42,58)(44,55)(45,62)(47,67)(48,80)(49,64)(51,61)(53,63)(54,79)(60,72)(65,73)(66,75)(68,71)(70,81)(74,82)(76,78)(77,83);(20,22,28,42)(21,29,35,59)(23,44,48,74)(24,41,27,38)(25,40,52,73)(26,39,49,75)(30,60,63,69)(31,58,34,36)(32,65,64,57)(33,66,50,56)(37,51,72,47)(43,78,80,68)(45,71,54,76)(46,70,53,77)(55,79,82,62)(61,83,67,81);(20,23,24,45,28,48,27,54)(21,30,31,61,35,63,34,67)(22,37,38,70,42,72,41,77)(25,51,49,46,52,47,26,53)(29,55,36,68,59,82,58,78)(32,62,33,43,64,79,50,80)(39,74,73,71,75,44,40,76)(56,60,57,81,66,69,65,83)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6406);