Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C8×C9 · Schur multiplier (H2) = C3 |
| direct product | C3×#504⋊C9 |
| unambiguous semidirect product | (C3×#504)⋊C9 |
| Sylow | #504 ⋊ C3×C9 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C32×C16 | A4 |
| commutator, hypocenter | Q8 | C3×C8×C9 |
| Frattini | C3×C8 | C2×C3×A4 |
| Fitting | C32×#504 | C3 |
| socle | C2×C32 | C8×A4 |
| prime radical | C32×#41 | C3×C4 |
| cosocle | C3×#131 | C2×C32 |
| LCS | G ▷ Q8 | |
| UCS | C1 ◁ C32×C16 | |
156 conjugacy classes of subgroups.
| transitive degree | μt(G) = 864 |
504 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 7328);