Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8×C19 · Schur multiplier (H2) = C23 |
| direct product | C19×#8604 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | #121 |
| commutator | C2×C4 | C22×C8×C19 |
| Frattini, cosocle | C2×C42 | C23×C19 |
| socle | C22×C19 | #380 |
| prime radical | C2×#41×C19 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C2×C42×C19 ◁ G | |
462 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)(28,29,33,36,34,31,32,42)(30,37,40,38,41,43,39,35);(28,30,32,39,34,41,33,40)(29,35,36,37,31,38,42,43);(28,31)(29,34)(30,38)(32,42)(33,36)(35,41)(37,40)(39,43)1216 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 1592);