Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12741 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#12742 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C2×C19 | #1786 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C2×C19×#578 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C19 ◁ C22×C4×C19 ◁ C2×C19×#131 ◁ G | |
878 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)(22,32)(23,37,39,28)(24,41)(25,46)(26,30)(27,34)(29,44)(31,43)(33,59,61,57)(35,65)(36,58)(38,69,72,75)(40,52,71,68)(42,55)(45,56)(47,53)(48,60,66,82)(49,64)(50,76)(51,73,67,81)(54,63)(62,78,74,80)(70,77,83,79);(20,22,26,36)(21,27,30,49)(23,33,39,61)(24,42,45,76)(25,47,43,35)(28,48,37,66)(29,53,54,65)(31,55,46,50)(32,56,58,41)(34,63,64,44)(38,70,72,83)(40,74,71,62)(51,78,67,80)(52,79,68,77)(57,69,59,75)(60,81,82,73);(20,23)(21,28)(22,33)(24,38)(25,40)(26,39)(27,48)(29,51)(30,37)(31,52)(32,57)(34,60)(35,62)(36,61)(41,75)(42,70)(43,71)(44,73)(45,72)(46,68)(47,74)(49,66)(50,77)(53,78)(54,67)(55,79)(56,69)(58,59)(63,81)(64,82)(65,80)(76,83)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5730);