Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C32 · Schur multiplier (H2) = C23×C3 |
| direct product | C3×#5964⋊C3 |
| unambiguous semidirect product | #5964⋊C32 |
| unambiguous semidirect product | (C3×#5964)⋊C3 |
| Fitting | C2×C32×#41 ⋊ C2×C3 |
| subgroup | G/H | |
|---|---|---|
| center | C3×C4 | C2×D3×A4 |
| commutator, cosocle, hypocenter | C3×Q8 | C23×C32 |
| Frattini | C2 | C22×C3×D3×A4 |
| Fitting | C2×C32×#41 | C2×C3 |
| socle | C2×C32 | C23×A4 |
| hypercenter | C2×C3×C4 | D3×A4 |
| LCS | G ▷ C3×Q8 | |
| UCS | C1 ◁ C3×C4 ◁ C2×C3×C4 | |
1051 conjugacy classes of subgroups.
| transitive degree | μt(G) = 96 |
198 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47512);