Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#12613 |
| subgroup | G/H | |
|---|---|---|
| center | C16×C19 | C22⋊C4 |
| commutator | C22 | C2×C4×C8×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C2×C19 | C8×C22⋊C4 |
| prime radical | C2×#41×C19 | C2×C4 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16×C19 ◁ C22×C16×C19 ◁ G | |
410 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)(20,21,24,29,25,31,40,51,26,30,41,50,42,52,64,73)(22,32,35,54,36,55,60,76,34,27,43,45,59,46,66,69)(23,28,37,47,38,49,61,71,39,48,62,70,63,72,79,82)(33,53,57,74,58,75,78,83,56,44,65,67,77,68,80,81);(20,22)(21,27,30,32)(23,33)(24,35)(25,36)(26,34)(28,44,48,53)(29,45,50,54)(31,46,52,55)(37,57)(38,58)(39,56)(40,60)(41,43)(42,59)(47,67,70,74)(49,68,72,75)(51,69,73,76)(61,78)(62,65)(63,77)(64,66)(71,81,82,83)(79,80);(20,23,24,37,25,38,40,61,26,39,41,62,42,63,64,79)(21,28,29,47,31,49,51,71,30,48,50,70,52,72,73,82)(22,33,35,57,36,58,60,78,34,56,43,65,59,77,66,80)(27,44,45,67,46,68,69,81,32,53,54,74,55,75,76,83)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5601);