Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8626 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#8635 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #380 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C2×C42 | C23×C19 |
| prime radical | C23×C19 | #117 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C2×C42×C19 ◁ G | |
320 conjugacy classes of subgroups.
| minimal degree | μ(G) = 67 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 3 generators on 67 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,24,27,23,25,31,33)(22,28,30,35,29,26,32,34)(36,37)(38,47)(39,52,54,59)(40,55)(41,45)(42,46)(43,49)(44,53,60,56)(48,57,64,67)(50,58)(51,62)(61,63,66,65);(21,25)(26,28)(27,33)(34,35)(36,38,40,49,42,51,41,50)(37,43,45,47,46,58,55,62)(39,48,53,63,54,64,56,65)(44,57,59,61,60,67,52,66);(20,22,23,29)(21,26,25,28)(24,32,31,30)(27,35,33,34)(36,39)(37,44)(38,48)(40,53)(41,56)(42,54)(43,57)(45,59)(46,60)(47,61)(49,63)(50,65)(51,64)(52,55)(58,67)(62,66)760 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 1623);