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×#33859 |
| 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)(35,55,41,79)(36,54,76,58)(38,77,52,61)(42,80,74,59)(46,66,70,56)(47,62,81,83)(67,73,82,72)(68,75,78,71);(20,22,27,39)(21,28,33,57)(23,41,44,35)(24,46,50,70)(25,38,26,52)(29,58,60,54)(30,61,64,77)(31,56,32,66)(34,63,69,45)(36,74,76,42)(37,48,53,49)(40,78,65,68)(43,73,51,72)(47,75,81,71)(55,80,79,59)(62,82,83,67);(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)(46,76)(47,49)(48,81)(50,74)(52,72)(53,75)(55,77)(56,78)(57,58)(61,79)(63,83)(64,80)(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)(38,75)(39,76)(41,46)(42,49)(45,59)(48,74)(50,81)(52,71)(53,73)(54,77)(56,82)(57,79)(58,61)(63,80)(64,83)(66,67)(69,78)817 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26847);