Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33539 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C26 |
| direct product | C19×#33542 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | C2×#447 |
| commutator, Frattini, cosocle | C2×C8 | C24×C19 |
| prime radical | C19×#3584 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C22×C4×C19 ◁ C2×C19×#131 ◁ G | |
1084 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)(22,34,38,62)(23,39,43,54)(24,32,26,30)(25,45,48,57)(28,50,53,37)(29,40,56,42)(31,44,59,47)(35,63,67,81)(36,68,49,58)(41,72,75,55)(46,60,76,52)(51,73,79,83)(61,64,82,66)(65,69,77,78)(70,71,80,74);(20,22)(21,28)(23,35)(24,36)(25,46)(26,49)(27,38)(29,51)(30,52)(31,58)(32,60)(33,53)(34,45)(37,47)(39,69)(40,64)(41,73)(42,66)(43,67)(44,50)(48,76)(54,78)(55,81)(56,79)(57,62)(59,68)(61,70)(63,72)(65,74)(71,77)(75,83)(80,82);(20,23)(21,29)(22,35)(24,40)(25,41)(26,42)(27,43)(28,51)(30,54)(31,55)(32,39)(33,56)(34,61)(36,64)(37,65)(38,67)(44,71)(45,70)(46,73)(47,74)(48,75)(49,66)(50,77)(52,78)(53,79)(57,80)(58,81)(59,72)(60,69)(62,82)(63,68)(76,83);(20,24,27,26)(21,30,33,32)(22,36,38,49)(23,40,43,42)(25,44,48,47)(28,52,53,60)(29,54,56,39)(31,57,59,45)(34,58,62,68)(35,64,67,66)(37,46,50,76)(41,71,75,74)(51,78,79,69)(55,80,72,70)(61,81,82,63)(65,73,77,83)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26530);