Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13442 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#13443 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #2054 |
| 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 | |
378 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)(20,21,27,33)(22,34,39,69)(23,40,46,76)(24,31,49,66)(25,30,50,65)(26,51,48,32)(28,53,57,36)(29,58,63,43)(35,70,74,54)(37,56,47,67)(38,55,52,64)(41,77,71,82)(42,62,44,80)(45,60,79,61)(59,72,68,81)(73,75,78,83);(20,22)(21,28)(23,41)(24,47)(25,36)(26,52)(27,39)(29,59)(30,64)(31,34)(32,67)(33,57)(35,44)(37,49)(38,48)(40,75)(42,74)(43,73)(45,81)(46,71)(50,53)(51,56)(54,61)(55,65)(58,78)(60,70)(62,77)(63,68)(66,69)(72,79)(76,83)(80,82);(20,23,24,42,48,79,50,58,27,46,49,44,26,45,25,43)(21,29,30,60,51,80,66,76,33,63,65,61,32,62,31,40)(22,35,36,71,52,78,37,72,39,74,53,41,38,73,47,81)(28,54,34,68,67,83,55,82,57,70,69,59,56,75,64,77)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6431);