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×#33860 |
| 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 | |
636 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)(20,21,28,34)(22,35,40,70)(23,41,30,56)(24,45,48,44)(25,49,54,65)(26,33,27,32)(29,57,61,38)(31,52,66,53)(36,37,42,75)(39,77,55,63)(43,73,62,47)(46,80,58,59)(50,67,82,60)(51,64,83,81)(68,69,74,72)(71,76,79,78);(20,22,28,40)(21,29,34,61)(23,42,30,36)(24,46,48,58)(25,50,54,82)(26,39,27,55)(31,63,66,77)(32,60,33,67)(35,65,70,49)(37,73,75,47)(38,52,57,53)(41,78,56,76)(43,80,62,59)(44,69,45,72)(51,74,83,68)(64,71,81,79);(20,23,28,30)(21,24,34,48)(22,36,40,42)(25,43,54,62)(26,45,27,44)(29,58,61,46)(31,47,66,73)(32,56,33,41)(35,68,70,74)(37,63,75,77)(38,71,57,79)(39,69,55,72)(49,51,65,83)(50,59,82,80)(52,64,53,81)(60,78,67,76);(20,24)(21,30)(22,37)(23,34)(25,51)(26,41)(27,56)(28,48)(29,59)(31,64)(32,45)(33,44)(35,69)(36,63)(38,76)(39,74)(40,75)(42,77)(43,65)(46,50)(47,53)(49,62)(52,73)(54,83)(55,68)(57,78)(58,82)(60,71)(61,80)(66,81)(67,79)(70,72)817 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26848);