Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C25 |
| direct product | C19×#8319 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #120 |
| commutator | C2×C4 | C22×C8×C19 |
| Frattini, cosocle | C2×C42 | C23×C19 |
| socle | C22×C19 | #379 |
| prime radical | C2×C19×2−1+4 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C42×C19 ◁ G | |
954 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)(20,21,22,24,23,25,26,27)(29,33)(31,38)(34,42)(35,39)(36,40)(41,43);(28,29)(30,34)(31,39)(32,35)(33,38)(36,41)(37,43)(40,42);(28,30,32,37)(29,34,35,43)(31,40,38,36)(33,41,39,42)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 1307);