Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
| direct product | D3×A4×#67 |
| unambiguous semidirect product | (C22×D3×#67)⋊C3 |
| Fitting | C24×C32 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C2×D32×A4 |
| commutator, socle, cosocle | C23×C32 | C23×C3 |
| Fitting | C24×C32 | C22×C3 |
| hypercenter | C22 | D32×A4 |
| hypocenter | C22×C32 | C2×C3×D4 |
| LCS | G ▷ C23×C32 ▷ C22×C32 | |
| UCS | C1 ◁ C2 ◁ C22 | |
1450 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
108 conjugacy classes — view the full character table.
| maximal subgroup of | D3×A4×C3⋊S4, D32×D4×A4, D3×#67×S4, … |
| maximal normal subgroup of | D32×D4×A4, D3×#67×S4, D3×#10596473 |
| maximal quotient of | D3×A4×GL(2,ℤ/4) |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46978);