Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2221 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#13144 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #466 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C22×D8×C19 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ C19×#405 ◁ G | |
334 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)(20,21,23,22)(25,30)(26,35)(27,31)(28,39)(32,47)(33,50)(34,44)(36,38)(37,45)(40,46)(41,48)(42,43)(49,53)(51,52);(20,22,23,21)(24,25,31,43,28,33,40,44,29,34,46,50,39,42,27,30)(26,32,45,54,37,47,35,48,38,49,52,55,51,53,36,41);(24,26,27,36,39,51,46,52,29,38,40,35,28,37,31,45)(25,32,30,41,42,53,50,55,34,49,44,48,33,47,43,54)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6132);