Finite Group DatabasebetaFinite Group Databasebeta| order | 2916 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C32 · Schur multiplier (H2) = C2×C3 |
| direct product | C3×(#4174⋊C2)⋊C3 |
| Fitting, Sylow | C3×#6909 ⋊ C22 |
| derived | AΣL(1,𝔽27) ⋊ C22×C32 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C3 | (#4174⋊C2)⋊C3 |
| commutator, cosocle, hypocenter | AΣL(1,𝔽27) | C22×C32 |
| Frattini | C32 | C32×D32 |
| Fitting | C3×#6909 | C22 |
| socle | C32 | C3×((C3⋊D3)⋊C3)⋊C2 |
| LCS | G ▷ AΣL(1,𝔽27) | |
| UCS | C1 ◁ C3 | |
671 conjugacy classes of subgroups; only the 68 normal subgroups are shown.
This group has 671 conjugacy classes of subgroups — too many to draw. Only its 68 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 54 |
105 conjugacy classes — view the full character table.
| maximal subgroup of | #72243811, C3×#72221953, C3×#72243770, … |
| maximal normal subgroup of | D3×(#4174⋊C2)⋊C3 |
Past the SmallGroups library coverage — no SmallGroups id.