Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C26 |
| direct product | D4×#611⋊C3 |
| unambiguous semidirect product | (D4×#611)⋊C3 |
| Fitting | C22×D4×3−1+2 ⋊ C2 |
| Sylow | 3−1+2 ⋊ D42 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×AGL(1,ℤ/9) |
| commutator, cosocle | C22×C9 | C24×C3 |
| Frattini, socle | C22×C3 | C23×C3×D3 |
| Fitting | C22×D4×3−1+2 | C2 |
| hypercenter | C22×D4 | AGL(1,ℤ/9) |
| hypocenter | C9 | C3×D42 |
| LCS | G ▷ C22×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ C22×D4 | |
1402 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
155 conjugacy classes — view the full character table.
| maximal subgroup of | D42×AGL(1,ℤ/9), #72268828, #72268843, … |
| maximal normal subgroup of | D42×AGL(1,ℤ/9), #72268828, #72268843, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 19690);