Finite Group DatabasebetaFinite Group Databasebeta| order | 6804 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | AGL(1,𝔽7)×D27⋊C3 |
| Fitting | C7×C27⋊C3 ⋊ C22×C3 |
| derived | C7×C27 ⋊ C22×C32 |
| Hall | ΓL(1,𝔽8)×C27⋊C3 ⋊ C22 |
| Sylow | C7 ⋊ C2×C3×D27⋊C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C7×C27 | C22×C32 |
| Frattini | C9 | C3×D3×AGL(1,𝔽7) |
| Fitting | C7×C27⋊C3 | C22×C3 |
| socle | C3×C7 | C2×C32×D9 |
| LCS | G ▷ C7×C27 | |
| UCS | C1 | |
312 conjugacy classes of subgroups; only the 67 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 67 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 189 |
147 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,𝔽7)×AGL(1,ℤ/27) |
| maximal normal subgroup of | AGL(1,𝔽7)×AGL(1,ℤ/27) |
Past the SmallGroups library coverage — no SmallGroups id.