Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#11599 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #102 |
| commutator | C8 | C22×C8×C19 |
| Frattini, cosocle | C4×C8 | C23×C19 |
| socle | C22×C19 | C2×#102 |
| prime radical | C22×D8×C19 | C4 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C42×C19 ◁ C2×C4×C8×C19 ◁ G | |
450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 43 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,22,24,23,25,26,27)(28,29,30,34,32,37,38,36)(31,40,39,35,42,33,41,43);(29,33)(30,38)(31,41)(34,35)(36,43)(37,40)(39,42);(28,30,32,38)(29,34,37,36)(31,39,42,41)(33,43,40,35)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 4587);