Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11669 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#12746 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #418 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | #1786 |
| prime radical | C19×#3584 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C2×C19×#131 ◁ G | |
886 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)(20,21,25,32,27,31,26,49)(22,33,37,62,36,63,50,58)(23,38,40,57,41,29,48,56)(24,42,45,60,47,74,46,30)(28,43,52,69,54,76,53,35)(34,61,66,83,65,79,67,82)(39,71,73,59,44,77,78,55)(51,72,80,64,70,75,81,68);(20,22)(21,28)(23,34)(24,43)(25,37)(26,50)(27,36)(29,51)(30,58)(31,54)(32,52)(33,42)(35,46)(38,70)(39,72)(40,66)(41,65)(44,75)(45,69)(47,76)(48,67)(49,53)(55,82)(56,80)(57,81)(59,83)(60,62)(61,71)(63,74)(64,73)(68,78)(77,79);(20,23)(21,29)(22,34)(24,44)(25,48)(26,40)(27,41)(28,51)(30,59)(31,38)(32,57)(33,61)(35,68)(36,65)(37,67)(39,47)(42,71)(43,75)(45,73)(46,78)(49,56)(50,66)(52,81)(53,80)(54,70)(55,60)(58,83)(62,82)(63,79)(64,69)(72,76)(74,77)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5734);