Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of D3×(C3⋊D3)⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26×C3 |
| direct product | C22×D3×(C3⋊D3)⋊C3 |
| Fitting | C22×C3×H(3,𝔽3) ⋊ C22 |
| derived | C33 ⋊ C24×C3 |
| Sylow | C3×H(3,𝔽3) ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | D3×(C3⋊D3)⋊C3 |
| commutator, cosocle, hypocenter | C33 | C24×C3 |
| Frattini | C3 | C22×C3×D32 |
| Fitting | C22×C3×H(3,𝔽3) | C22 |
| socle | C22×C32 | C2×C3×D3 |
| LCS | G ▷ C33 | |
| UCS | C1 ◁ C22 | |
1166 conjugacy classes of subgroups; only the 229 normal subgroups are shown.
This group has 1166 conjugacy classes of subgroups — too many to draw. Only its 229 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
120 conjugacy classes — view the full character table.
| maximal subgroup of | C22×C3⋊D3×(C3⋊D3)⋊C3, C22×D3×PSU(3,𝔽2)⋊C3, C2×D32×(C3⋊D3)⋊C3, … |
| maximal normal subgroup of | C22×D3×((C3⋊D3)⋊C3)⋊C2 |
| maximal quotient of | C22×C3⋊D3×(C3⋊D3)⋊C3, C2×D32×(C3⋊D3)⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3032);