Finite Group DatabasebetaFinite Group Databasebeta| order | 4864 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4×C19 · Schur multiplier (H2) = C2×C4 |
| direct product | C19×#13154 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C22×C19 | #442 |
| commutator | C2×C8 | C22×C4×C19 |
| Frattini, cosocle | C22×C8 | C23×C19 |
| prime radical | C23×C19 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C19 ◁ C22×C4×C19 ◁ C2×C19×#97 ◁ G | |
314 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)(24,25,29,34,30,36,48,35)(26,37,39,32,41,50,40,51)(27,42,43,54,45,31,44,49)(28,46,47,33,38,52,55,53);(24,26,38,27,29,40,55,44,30,41,28,45,48,39,47,43)(25,31,33,50,35,49,46,51,36,42,53,37,34,54,52,32);(21,23)(24,27,28,40,29,44,47,41,30,45,38,39,48,43,55,26)(25,32,33,42,35,50,46,54,36,51,53,31,34,37,52,49)646 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(4864, 6142);