Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33657 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#33658 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | C2×#481 |
| commutator, Frattini, cosocle | C2×C8 | C24×C19 |
| prime radical | C2×#41×C19 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C22×C4×C19 ◁ C2×C19×#131 ◁ G | |
534 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)(20,21,26,31)(22,27,33,49)(23,36,37,71)(24,40,45,76)(25,46,43,30)(28,52,53,38)(29,55,58,44)(32,60,61,70)(34,65,41,75)(35,68,64,51)(39,57,47,72)(42,54,74,59)(48,73,79,62)(50,78,56,67)(63,81,69,83)(66,80,77,82);(20,22)(21,27)(23,32)(24,41)(25,35)(26,33)(28,48)(29,56)(30,51)(31,49)(34,45)(36,70)(37,61)(38,73)(39,63)(40,75)(42,77)(43,64)(44,78)(46,68)(47,69)(50,58)(52,62)(53,79)(54,80)(55,67)(57,83)(59,82)(60,71)(65,76)(66,74)(72,81);(20,23,25,47,26,37,43,39)(21,28,30,59,31,53,46,54)(22,32,35,69,33,61,64,63)(24,42,44,52,45,74,55,38)(27,48,51,82,49,79,68,80)(29,57,40,71,58,72,76,36)(34,66,67,73,41,77,78,62)(50,81,65,70,56,83,75,60);(22,33)(27,49)(32,61)(34,41)(35,64)(48,79)(50,56)(51,68)(60,70)(62,73)(63,69)(65,75)(66,77)(67,78)(80,82)(81,83)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 26646);