Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#33442 |
| subgroup | G/H | |
|---|---|---|
| center | C8×C19 | C22×D4 |
| commutator | C4 | C23×C8×C19 |
| Frattini, cosocle | C2×C8 | C24×C19 |
| socle | C2×C19 | C2×C8×D4 |
| prime radical | C19×#575 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8×C19 ◁ C2×C19×M32 ◁ G | |
642 conjugacy classes of subgroups.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 4 generators on 83 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,25,31,26,30,45,55,27,32,46,56,24,42,44,68)(22,33,36,58,35,28,47,49,37,50,62,74,43,51,69,75)(23,29,40,54,41,53,67,77,38,52,63,76,39,64,66,80)(34,57,61,78,60,48,70,72,59,73,79,83,65,71,81,82);(20,22)(21,28)(23,34)(24,43)(25,36)(26,35)(27,37)(29,48)(30,50)(31,49)(32,51)(33,42)(38,59)(39,65)(40,61)(41,60)(44,69)(45,47)(46,62)(52,71)(53,73)(54,72)(55,74)(56,75)(57,64)(58,68)(63,79)(66,81)(67,70)(76,82)(77,83)(78,80);(20,23,25,40,26,41,45,67,27,38,46,63,24,39,44,66)(21,29,31,54,30,53,55,77,32,52,56,76,42,64,68,80)(22,34,36,61,35,60,47,70,37,59,62,79,43,65,69,81)(28,48,49,72,50,73,74,83,51,71,75,82,33,57,58,78);(23,38)(29,52)(34,59)(39,41)(40,63)(48,71)(53,64)(54,76)(57,73)(60,65)(61,79)(66,67)(70,81)(72,82)(77,80)(78,83)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26430);