Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#33617 |
| subgroup | G/H | |
|---|---|---|
| center | C4×C19 | #466 |
| commutator, Frattini, cosocle | C2×C8 | C24×C19 |
| socle | C2×C19 | C2×#466 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4×C19 ◁ C22×C4×C19 ◁ C2×C19×#131 ◁ 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)(22,34)(23,41)(24,30)(25,47)(26,32)(27,33)(28,38)(29,44)(31,35)(36,64)(37,59)(39,60)(40,61)(42,54)(43,74)(45,57)(46,56)(48,55)(49,76)(50,69)(51,62)(52,70)(53,75)(58,67)(63,72)(65,80)(66,71)(68,81)(73,79)(77,82)(78,83);(20,22,35,62,26,39,48,70,27,40,46,69,24,37,25,38)(21,28,47,59,30,50,56,61,33,52,55,60,32,51,31,34)(23,36,63,76,44,67,75,83,45,68,73,82,42,65,43,66)(29,49,72,64,41,71,74,80,54,77,79,81,57,78,53,58);(20,23)(21,29)(22,36)(24,42)(25,43)(26,44)(27,45)(28,49)(30,41)(31,53)(32,57)(33,54)(34,58)(35,63)(37,65)(38,66)(39,67)(40,68)(46,73)(47,72)(48,75)(50,71)(51,78)(52,77)(55,79)(56,74)(59,64)(60,81)(61,80)(62,76)(69,82)(70,83);(20,24,27,26)(21,30,33,32)(22,37,40,39)(23,42,45,44)(25,46,48,35)(28,50,52,51)(29,41,54,57)(31,47,56,55)(34,59,61,60)(36,65,68,67)(38,69,70,62)(43,73,75,63)(49,71,77,78)(53,72,74,79)(58,64,80,81)(66,82,83,76)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26605);