Finite Group DatabasebetaFinite Group Databasebeta| order | 3240 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32×C5 · Schur multiplier (H2) = C2×C3 |
| direct product | C5×A4×AGL(1,ℤ/9) |
| Fitting | C22×C5×3−1+2 ⋊ C2×C3 |
| derived | C22×C9 ⋊ C2×C32×C5 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C5 | A4×AGL(1,ℤ/9) |
| commutator, cosocle, hypocenter | C22×C9 | C2×C32×C5 |
| Frattini | C3 | C3×C5×D3×A4 |
| Fitting | C22×C5×3−1+2 | C2×C3 |
| socle | C22×C3×C5 | C32×D3 |
| LCS | G ▷ C22×C9 | |
| UCS | C1 ◁ C5 | |
234 conjugacy classes of subgroups; only the 54 normal subgroups are shown.
This group has 234 conjugacy classes of subgroups — too many to draw. Only its 54 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 180 |
200 conjugacy classes — view the full character table.
| maximal subgroup of | D5×A4×AGL(1,ℤ/9), C5×S4×AGL(1,ℤ/9), AGL(1,ℤ/9)×C5⋊S4 |
| maximal normal subgroup of | D5×A4×AGL(1,ℤ/9), C5×S4×AGL(1,ℤ/9), AGL(1,ℤ/9)×C5⋊S4 |
Past the SmallGroups library coverage — no SmallGroups id.