Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33856 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#33857 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | D4×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24×C19 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C2×C4×C19 ◁ C19×#443 ◁ G | |
646 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)(22,34,39,69)(23,40,44,65)(24,45,50,63)(25,32,26,31)(28,53,57,37)(29,51,60,43)(30,48,64,49)(35,55,41,78)(36,54,75,58)(38,76,52,61)(42,79,80,59)(46,66,70,56)(47,62,81,83)(67,73,82,72)(68,74,77,71);(20,22)(21,28)(23,41)(24,46)(25,38)(26,52)(27,39)(29,58)(30,61)(31,56)(32,66)(33,57)(34,45)(35,44)(36,42)(37,49)(40,77)(43,73)(47,74)(48,53)(50,70)(51,72)(54,60)(55,59)(62,82)(63,69)(64,76)(65,68)(67,83)(71,81)(75,80)(78,79);(20,23)(21,29)(22,35)(24,42)(25,51)(26,43)(27,44)(28,54)(30,59)(31,65)(32,40)(33,60)(34,67)(36,70)(37,71)(38,73)(39,41)(45,62)(46,75)(47,49)(48,81)(50,80)(52,72)(53,74)(55,76)(56,77)(57,58)(61,78)(63,83)(64,79)(66,68)(69,82);(20,23)(21,29)(22,36)(24,47)(25,43)(26,51)(27,44)(28,55)(30,62)(31,40)(32,65)(33,60)(34,68)(35,70)(37,72)(38,74)(39,75)(41,46)(42,49)(45,59)(48,80)(50,81)(52,71)(53,73)(54,76)(56,82)(57,78)(58,61)(63,79)(64,83)(66,67)(69,77)817 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26845);