Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#8171 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C19 | C2×D4 |
| commutator | C4 | C2×C4×C8×C19 |
| Frattini, cosocle | C2×C42 | C23×C19 |
| socle, prime radical | C23×C19 | C2×C22⋊C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C19 ◁ C43×C19 ◁ G | |
598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 39 |
| transitive degree | μt(G) = 4864 |
A minimal faithful permutation representation — 3 generators on 39 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,22)(32,33)(34,35)(36,38)(37,39);(24,25,26,28,27,29,30,31)(32,34,36,39,37,38,35,33);(20,22,23,21)(32,35,37,36)(33,34,39,38)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 1159);