Finite Group DatabasebetaFinite Group Databasebeta| order | 2880 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
| direct product | C4×PGL(2,𝔽9) |
| derived | A6 ⋊ C2×C4 |
| subgroup | G/H | |
|---|---|---|
| center, Fitting, hypercenter | C4 | PGL(2,𝔽9) |
| commutator, hypocenter | A6 | C2×C4 |
| Frattini | C2 | C2×PGL(2,𝔽9) |
| socle, cosocle | C2×A6 | C22 |
| prime radical | C2×PGL(2,𝔽9) | C2 |
| LCS | G ▷ A6 | |
| UCS | C1 ◁ C4 | |
136 conjugacy classes of subgroups; only the 11 normal subgroups are shown.
This group has 136 conjugacy classes of subgroups — too many to draw. Only its 11 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 14 |
| transitive degree | μt(G) = 40 |
A minimal faithful permutation representation — 2 generators on 14 points:
(1,3,2,4)(5,13,11,10,6)(7,12,8,14,9);(5,8,14,10,6,12,7,13)44 conjugacy classes — view the full character table.
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((1,3,2,4)(5,13,11,10,6)(7,12,8,14,9), (5,8,14,10,6,12,7,13));