Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C32 · Schur multiplier (H2) = C29×C3 |
| direct product | C32×#5968 |
| Fitting | C33×2−1+4 ⋊ C2 |
| Sylow | C33 ⋊ #585 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C32 | C24×D3 |
| commutator, cosocle | C2×C3 | C25×C32 |
| Frattini | C2 | C24×C32×D3 |
| Fitting | C33×2−1+4 | C2 |
| socle | C2×C33 | C25 |
| hypercenter | C32×2−1+4 | D3 |
| hypocenter | C3 | C32×#585 |
| LCS | G ▷ C2×C3 ▷ C3 | |
| UCS | C1 ◁ C2×C32 ◁ C32×2−1+4 | |
5444 conjugacy classes of subgroups.
| transitive degree | μt(G) = 288 |
459 conjugacy classes. The character table is too large to display.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47790);