Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| isoclinism | isoclinic class of AGL(1,ℤ/9) |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C9 · Schur multiplier (H2) = C26 |
| direct product | C23×D9⋊C9 |
| unambiguous semidirect product | D9⋊(C23×C9) |
| unambiguous semidirect product | (C9⋊C9)⋊C24 |
| unambiguous semidirect product | (C2×C9⋊C9)⋊C23 |
| Fitting | C23×C9⋊C9 ⋊ C2 |
| derived | C9 ⋊ C24×C9 |
| Sylow | C9⋊C9 ⋊ C24 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3 | AGL(1,ℤ/9) |
| commutator, hypocenter | C9 | C24×C9 |
| Frattini | C32 | C23×C3×D3 |
| Fitting | C23×C9⋊C9 | C2 |
| socle | C23×C32 | C3×D3 |
| prime radical | C23×C3×D9 | C3 |
| cosocle | C3×C9 | C24×C3 |
| LCS | G ▷ C9 | |
| UCS | C1 ◁ C23×C3 | |
667 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
240 conjugacy classes — view the full character table.
| maximal subgroup of | C23×#92265, #72220016, C2×D4×D9⋊C9, … |
| maximal normal subgroup of | C23×#92265, #72220016, C2×D4×D9⋊C9 |
| maximal quotient of | #72219942, C2×D4×D9⋊C9, C23×AGL(1,ℤ/27) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 1904);