Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#19093 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24×C19 |
| prime radical | C19×#121 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C2×C42×C19 ◁ G | |
782 conjugacy classes of subgroups.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 2432 |
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)(20,21,26,31)(22,32,34,44)(23,36,38,45)(24,30,25,29)(27,39,42,35)(28,40,43,37)(33,41,47,51)(46,48,50,49)(52,53,57,60)(54,62,64,76)(55,59,67,74)(56,68,71,61)(58,69,73,65)(63,79,80,78)(66,77,70,75)(72,82,83,81);(20,22)(21,27)(23,33)(24,39)(25,35)(26,34)(28,41)(29,44)(30,32)(31,42)(36,50)(37,49)(38,47)(40,48)(43,51)(45,46)(52,54,57,64)(53,58,60,73)(55,63,67,80)(56,69,71,65)(59,72,74,83)(61,76,68,62)(66,82,70,81)(75,78,77,79);(20,23,26,38)(21,28,31,43)(22,33,34,47)(24,37,25,40)(27,41,42,51)(29,36,30,45)(32,46,44,50)(35,48,39,49)(52,55,57,67)(53,59,60,74)(54,63,64,80)(56,70,71,66)(58,72,73,83)(61,77,68,75)(62,78,76,79)(65,82,69,81);(22,34)(27,42)(32,44)(33,47)(35,39)(41,51)(46,50)(48,49)(53,60)(54,64)(59,74)(61,68)(63,80)(65,69)(75,77)(81,82)589 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 12081);