Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33854 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#33855 |
| 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 | |
640 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)(22,34,39,67)(23,40,43,63)(24,44,49,61)(25,32,26,31)(28,52,56,37)(29,50,58,42)(30,47,62,48)(35,68,72,54)(36,73,76,53)(38,77,51,59)(41,79,74,57)(45,64,80,55)(46,60,81,83)(65,71,82,70)(66,69,78,75);(20,22,27,39)(21,28,33,56)(23,35,43,72)(24,45,49,80)(25,38,26,51)(29,53,58,73)(30,59,62,77)(31,55,32,64)(34,61,67,44)(36,74,76,41)(37,47,52,48)(40,78,63,66)(42,70,50,71)(46,69,81,75)(54,79,68,57)(60,65,83,82);(20,23)(21,29)(22,35)(24,41)(25,50)(26,42)(27,43)(28,53)(30,57)(31,63)(32,40)(33,58)(34,65)(36,45)(37,69)(38,71)(39,72)(44,60)(46,48)(47,81)(49,74)(51,70)(52,75)(54,59)(55,66)(56,73)(61,83)(62,79)(64,78)(67,82)(68,77)(76,80);(20,23)(21,29)(22,36)(24,46)(25,42)(26,50)(27,43)(28,54)(30,60)(31,40)(32,63)(33,58)(34,66)(35,45)(37,71)(38,69)(39,76)(41,48)(44,57)(47,74)(49,81)(51,75)(52,70)(53,59)(55,65)(56,68)(61,79)(62,83)(64,82)(67,78)(72,80)(73,77)703 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26843);