Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22×C4 |
| direct product | C19×#13241 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4×C19 | C2×D8 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| socle | C22×C19 | C4×D8 |
| prime radical | C2×C19×#508 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C22×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 55 |
| transitive degree | μt(G) = 1216 |
A minimal faithful permutation representation — 3 generators on 55 points:
(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)(21,23)(24,25,30,36,31,38,47,37)(26,33,41,53,42,50,55,54)(27,43,39,32,45,51,40,52)(28,46,48,34,29,49,44,35);(20,21,22,23)(24,26,29,40,47,55,48,45,31,42,28,39,30,41,44,27)(25,32,34,33,36,51,49,53,38,52,35,50,37,43,46,54);(24,27,44,41,30,39,28,42,31,45,48,55,47,40,29,26)(25,33,49,52,37,54,34,51,38,50,46,32,36,53,35,43)1444 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6229);