Finite Group DatabasebetaFinite Group Databasebeta| order | 3240 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| direct product | AGL(1,𝔽5)×D27⋊C3 |
| Fitting, Hall | C5×C27⋊C3 ⋊ C2×C4 |
| derived | C5×C27 ⋊ C2×C3×C4 |
| Sylow | C27⋊C3 ⋊ C2×AGL(1,𝔽5) |
| Sylow | C5 ⋊ C4×D27⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, hypocenter | C5×C27 | C2×C3×C4 |
| Frattini | C9 | C3×D3×AGL(1,𝔽5) |
| Fitting | C5×C27⋊C3 | C2×C4 |
| socle | C3×C5 | C3×C4×D9 |
| prime radical | D5×D27⋊C3 | C2 |
| cosocle | D5×C27 | C22×C3 |
| LCS | G ▷ C5×C27 | |
| UCS | C1 | |
140 conjugacy classes of subgroups; only the 40 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 40 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 135 |
105 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽5)×AGL(1,ℤ/27) |
| maximal normal subgroup of | AGL(1,𝔽5)×AGL(1,ℤ/27) |
Past the SmallGroups library coverage — no SmallGroups id.