Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8479 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#8490 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #379 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C2×C42 | C23×C19 |
| prime radical | C19×#538 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C2×C42×C19 ◁ G | |
630 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)(20,21,24,30,46,53,76,79)(22,32,35,61,34,67,44,75)(23,38,41,70,40,56,72,81)(25,28,45,52,26,31,47,59)(27,48,51,33,63,65,58,64)(29,54,57,43,60,74,83,42)(36,62,68,82,37,55,69,80)(39,49,71,77,66,50,73,78);(20,22,25,36)(21,27,28,49)(23,39,42,64)(24,44,45,69)(26,37,46,34)(29,55,38,67)(30,58,52,78)(31,50,53,63)(32,60,62,56)(33,40,66,43)(35,47,68,76)(41,73,54,65)(48,72,71,74)(51,59,77,79)(57,82,70,61)(75,83,80,81);(21,28)(23,40)(24,45)(27,49)(29,56)(31,53)(32,55)(33,64)(35,68)(38,60)(39,66)(41,74)(42,43)(44,69)(47,76)(48,73)(50,63)(54,72)(57,83)(61,75)(62,67)(65,71)(70,81)(80,82)703 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 1478);