Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#8224 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23×C19 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42×C19 |
| Frattini, cosocle | C2×C42 | C23×C19 |
| prime radical | C22×D4×C19 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C19 ◁ C43×C19 ◁ G | |
528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 47 |
| transitive degree | μt(G) = 2432 |
A minimal faithful permutation representation — 3 generators on 47 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(20,21,23,22)(25,28)(26,30)(32,33)(34,40)(35,43)(36,39)(37,42)(38,44)(41,46)(45,47);(24,25,26,29,27,28,30,31)(32,34)(33,37,39,45)(35,42)(36,41)(38,40,43,46)(44,47);(20,22,23,21)(34,41)(37,45)(40,46)(42,47)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 1212);