Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#13152 |
| 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×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C19 ◁ C22×C8×C19 ◁ C2×C4×C8×C19 ◁ G | |
324 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)(24,25,30,36,31,38,49,37)(26,39,43,52,27,44,47,55)(28,48,50,34,40,35,29,51)(32,46,33,45,54,42,53,41);(24,26,40,45,49,47,50,46,31,27,28,41,30,43,29,42)(25,32,34,39,36,33,35,52,38,54,51,44,37,53,48,55);(21,23)(24,27,28,45,49,43,29,46,31,26,40,41,30,47,50,42)(25,33,34,52,36,54,35,44,38,53,51,55,37,32,48,39)988 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6140);