Finite Group DatabasebetaFinite Group Databasebeta| order | 6480 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C22×C3 |
| direct product | D5×A4×AGL(1,ℤ/9) |
| Fitting | C22×C5×3−1+2 ⋊ C22×C3 |
| derived | C22×C5×C9 ⋊ C22×C32 |
| Sylow | C5 ⋊ C2×A4×AGL(1,ℤ/9) |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C22×C5×C9 | C22×C32 |
| Frattini | C3 | C3×D3×D5×A4 |
| Fitting | C22×C5×3−1+2 | C22×C3 |
| socle | C22×C3×C5 | C2×C32×D3 |
| LCS | G ▷ C22×C5×C9 | |
| UCS | C1 | |
676 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 676 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 180 |
A minimal faithful permutation representation — 2 generators on 18 points:
(2,3)(4,5)(6,10,14,8,7,9,13,12,11)(16,18,17);(1,2,4,5,3)(7,11,12,9,10,14)(8,13)(15,18)(16,17)160 conjugacy classes — view the full character table.
| maximal subgroup of | A4×AGL(1,𝔽5)×AGL(1,ℤ/9), D5×S4×AGL(1,ℤ/9), AGL(1,ℤ/9)×A4⋊AGL(1,𝔽5) |
| maximal normal subgroup of | A4×AGL(1,𝔽5)×AGL(1,ℤ/9), D5×S4×AGL(1,ℤ/9), AGL(1,ℤ/9)×A4⋊AGL(1,𝔽5) |
Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:
gap> G := Group((2,3)(4,5)(6,10,14,8,7,9,13,12,11)(16,18,17), (1,2,4,5,3)(7,11,12,9,10,14)(8,13)(15,18)(16,17));