Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#17876 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C22×D4 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C42 | C24×C19 |
| socle | C22×C19 | C2×C4×D4 |
| prime radical | C2×#41×C19 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C19×#431 ◁ G | |
650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 4 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,22,24,25,27,26)(28,29,33,37)(30,35,39,42)(31,36,32,41)(34,40,43,38);(28,30)(29,34)(31,38)(32,40)(33,39)(35,41)(36,42)(37,43);(20,22,23,25,24,26,27,21)(28,30)(29,35)(31,40)(32,38)(33,39)(34,41)(36,43)(37,42);(28,31,33,32)(29,36,37,41)(30,38,39,40)(34,42,43,35)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 10864);