Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8603 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C24×C4 |
| direct product | C19×#11098 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #117 |
| commutator | C22×C4 | C22×C4×C19 |
| Frattini, cosocle | C23×C4 | C23×C19 |
| socle | C22×C19 | #380 |
| prime radical | C22×#41×C19 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C22×M16×C19 ◁ G | |
882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(24,25)(26,34)(27,38,40,31)(28,41)(29,33)(30,36)(32,42)(35,48,50,46)(37,47)(39,45,54,53)(43,49,52,55)(44,51);(20,21)(22,23)(24,26,28,36,29,37,42,51)(25,30,32,34,33,44,41,47)(27,35,39,49,40,50,54,55)(31,43,45,46,38,52,53,48);(21,23)(24,27)(25,31)(26,35)(28,39)(29,40)(30,43)(32,45)(33,38)(34,46)(36,49)(37,50)(41,53)(42,54)(44,52)(47,48)(51,55)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4086);