Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C22 |
| direct product | C19×#13169 |
| 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 | C23×C19 | #135 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C22×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
286 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)(20,21)(22,23)(24,25,29,34,30,36,44,35)(26,31,28,43,38,33,45,47)(27,39,41,52,42,46,54,53)(32,48,49,55,51,37,50,40);(21,23)(24,26,29,28,30,38,44,45)(25,31,34,43,36,33,35,47)(27,37,41,40,42,48,54,55)(32,46,49,53,51,39,50,52);(24,27,38,48,29,41,45,55,30,42,26,37,44,54,28,40)(25,32,43,53,35,50,31,46,36,51,47,52,34,49,33,39)1444 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6157);