Finite Group DatabasebetaFinite Group Databasebeta| order | 1296 |
| automorphism | ? |
| homology | Abelianization (H1) = C33 · Schur multiplier (H2) = C24×C32 |
| direct product | 3−1+2×C22⋊A4 |
| Fitting | C24×3−1+2 ⋊ C3 |
| Sylow | C24 ⋊ C3×3−1+2 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C32×C22⋊A4 |
| commutator, socle, cosocle | C24×C3 | C33 |
| Fitting | C24×3−1+2 | C3 |
| prime radical | C32×C22⋊A4 | C3 |
| hypercenter | 3−1+2 | C22⋊A4 |
| hypocenter | C24 | C3×3−1+2 |
| LCS | G ▷ C24×C3 ▷ C24 | |
| UCS | C1 ◁ C3 ◁ 3−1+2 | |
426 conjugacy classes of subgroups.
| transitive degree | μt(G) = 108 |
88 conjugacy classes — view the full character table.
| maximal subgroup of | C22⋊A4×AGL(1,ℤ/9), 3−1+2×#1020, #72220104, … |
| maximal normal subgroup of | C22⋊A4×AGL(1,ℤ/9), 3−1+2×#1020, #72220104, … |
| maximal quotient of | 3−1+2×#997, C3×3−1+2×C22⋊A4, C22⋊A4×C9⋊C9, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1296, 3057);