Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24987 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#24989 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #2420 |
| commutator, Frattini, cosocle | C22×C4 | C24×C19 |
| prime radical | C2×#41×C19 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C22×C4×C19 ◁ G | |
620 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)(20,21,27,33)(22,34,39,52)(23,40,44,59)(24,45,48,57)(25,32,26,31)(28,38,53,37)(29,49,56,43)(30,47,58,46)(35,54,41,50)(36,64,67,51)(42,72,74,55)(60,68,69,63)(61,66,77,65)(62,81,71,80)(70,75,79,73)(76,82,78,83);(20,22,26,38,27,39,25,37)(21,28,32,52,33,53,31,34)(23,41,43,68,44,35,49,63)(24,36,47,66,48,67,46,65)(29,54,40,69,56,50,59,60)(30,51,57,77,58,64,45,61)(42,71,73,83,74,62,75,82)(55,78,70,81,72,76,79,80);(20,23)(21,29)(22,35)(24,42)(25,43)(26,49)(27,44)(28,50)(30,55)(31,40)(32,59)(33,56)(34,60)(36,62)(37,63)(38,68)(39,41)(45,70)(46,73)(47,75)(48,74)(51,76)(52,69)(53,54)(57,79)(58,72)(61,80)(64,78)(65,82)(66,83)(67,71)(77,81);(20,24,26,47,27,48,25,46)(21,30,32,57,33,58,31,45)(22,36,38,66,39,67,37,65)(23,42,49,75,44,74,43,73)(28,51,52,77,53,64,34,61)(29,55,59,79,56,72,40,70)(35,62,68,83,41,71,63,82)(50,76,69,81,54,78,60,80)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 17977);