Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33764 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C23×C4 |
| direct product | C19×#33789 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C19 |
| prime radical | C2×C19×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C19×#117 ◁ C19×#405 ◁ G | |
562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 91 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 4 generators on 91 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21)(22,26)(23,25)(24,27)(28,29,34,39)(30,35,41,56)(31,44,47,78)(32,48,52,77)(33,53,50,38)(36,59,61,45)(37,62,64,51)(40,66,67,89)(42,72,70,88)(43,75,71,58)(46,79,54,63)(49,65,80,60)(55,81,82,68)(57,85,84,74)(69,90,76,86)(73,87,91,83);(20,22,23,27)(21,24,25,26)(28,30)(29,35)(31,40)(32,42)(33,43)(34,41)(36,55)(37,57)(38,58)(39,56)(44,66)(45,68)(46,69)(47,67)(48,72)(49,73)(50,71)(51,74)(52,70)(53,75)(54,76)(59,81)(60,83)(61,82)(62,85)(63,86)(64,84)(65,87)(77,88)(78,89)(79,90)(80,91);(22,27)(24,26)(28,31,34,47)(29,36,39,61)(30,40,41,67)(32,49,52,80)(33,54,50,46)(35,55,56,82)(37,63,64,79)(38,65,53,60)(42,73,70,91)(43,76,71,69)(44,77,78,48)(45,62,59,51)(57,86,84,90)(58,87,75,83)(66,88,89,72)(68,85,81,74);(21,25)(24,26)(30,41)(35,56)(40,67)(42,70)(43,71)(55,82)(57,84)(58,75)(66,89)(68,81)(69,76)(72,88)(73,91)(74,85)(83,87)(86,90)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26777);