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×#8185 |
| subgroup | G/H | |
|---|---|---|
| center | C42×C19 | C2×D4 |
| commutator | C4 | C2×C4×C8×C19 |
| Frattini, cosocle | C2×C42 | C23×C19 |
| socle | C22×C19 | C4×C22⋊C4 |
| prime radical | C2×#41×C19 | C2×C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42×C19 ◁ C2×C4×C8×C19 ◁ G | |
514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 43 |
| transitive degree | μt(G) = 2432 |
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)(28,29,32,36,33,37,31,34)(30,35,40,42,38,43,39,41);(20,21,22,24,23,25,26,27)(29,34,37,36)(30,38)(35,42,43,41)(39,40);(28,30,32,40,33,38,31,39)(29,35,36,42,37,43,34,41)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 1173);