Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C3 · Schur multiplier (H2) = C29 |
| direct product | C3×#10593143 |
| Fitting | C3×C9×2−1+4 ⋊ C2 |
| Sylow | C3×C9 ⋊ #585 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C3 | C24×D9 |
| commutator, cosocle | C2×C9 | C25×C3 |
| Frattini | C2×C3 | C24×C3×D3 |
| Fitting | C3×C9×2−1+4 | C2 |
| socle | C2×C32 | C24×D3 |
| hypercenter | C3×2−1+4 | D9 |
| hypocenter | C9 | C3×#585 |
| LCS | G ▷ C2×C9 ▷ C9 | |
| UCS | C1 ◁ C2×C3 ◁ C3×2−1+4 | |
2576 conjugacy classes of subgroups.
| transitive degree | μt(G) = 288 |
306 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 31019);