Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4×C19 · Schur multiplier (H2) = C24 |
| direct product | C19×#17863 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C19 | C2×D4 |
| commutator | C2×C4 | C23×C4×C19 |
| Frattini, cosocle | C42 | C24×C19 |
| socle | C22×C19 | C2×C4×D4 |
| prime radical | C2×C19×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C19 ◁ C2×C4×C8×C19 ◁ G | |
688 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)(30,37)(31,35)(32,40)(33,36)(34,38)(39,42)(41,43);(28,30)(29,34)(31,38)(32,41)(33,39)(35,42)(36,43)(37,40);(20,22,23,25,24,26,27,21);(28,31,33,32)(29,35,36,40)(30,38,39,41)(34,42,43,37)1444 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 10851);