Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13068 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#13069 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1893 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×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 | |
450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 83 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 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)(22,33)(23,39,42,57)(24,43)(25,31)(26,49)(27,32)(28,36)(29,41,56,48)(30,45)(34,59)(35,66,69,82)(37,61)(38,63)(40,55,44,58)(46,50)(47,62)(51,78,81,83)(52,72)(53,64)(54,73)(60,68,79,70)(65,75,76,77)(67,74,71,80);(20,22,34,64,26,37,24,36,27,38,45,72,25,47,46,73)(21,28,50,61,31,53,30,33,32,54,43,62,49,52,59,63)(23,35,65,78,41,70,44,71,42,69,76,83,48,68,40,67)(29,51,77,82,57,80,58,60,56,81,75,66,39,74,55,79);(20,23)(21,29)(22,35)(24,44)(25,48)(26,41)(27,42)(28,51)(30,58)(31,57)(32,56)(33,60)(34,65)(36,71)(37,70)(38,69)(39,49)(40,46)(43,75)(45,76)(47,68)(50,77)(52,74)(53,80)(54,81)(55,59)(61,82)(62,66)(63,79)(64,78)(67,73)(72,83)874 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6057);