Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#8269 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C19 | C2×D4 |
| commutator | C2×C4 | C2×C42×C19 |
| Frattini, cosocle | C2×C42 | C23×C19 |
| socle | C22×C19 | C4×C4⋊C4 |
| prime radical | C23×C19 | #117 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C19 ◁ C2×C4×C8×C19 ◁ G | |
366 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)(29,33,35,38)(30,36)(34,41,42,40)(37,43);(20,21,22,25,23,24,26,27)(28,29)(30,34)(31,38)(32,35)(33,39)(36,42)(37,41)(40,43);(21,24)(25,27)(28,30,31,37,32,36,39,43)(29,34,38,41,35,42,33,40)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 1257);