Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42×C19 · Schur multiplier (H2) = C23×C4 |
| direct product | C4×C19×#357 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4×C19 | D8 |
| commutator | C8 | C2×C42×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C23×C19 | C2×D8 |
| prime radical | C22×D8×C19 | C22 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C19 ◁ C2×C42×C19 ◁ C2×C4×C8×C19 ◁ G | |
534 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)(25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43);(5,6,7,8)(9,10)(11,14)(12,21)(13,17)(15,18)(16,19)(20,23)(22,24);(10,14)(11,18)(12,19)(15,16)(17,23)(20,24)(21,22)1672 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 5623);