Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C29×C32 |
| unambiguous semidirect product | C32⋊(C22⋊(C22⋊A4)) |
| Fitting | C26×C32 ⋊ C3 |
| Sylow | C26 ⋊ H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C3 | C3×C22⋊(C22⋊A4) |
| commutator, socle, cosocle | C26×C3 | C32 |
| Fitting | C26×C32 | C3 |
| hypercenter | C32 | C22⋊(C22⋊A4) |
| hypocenter | C26 | H(3,𝔽3) |
| LCS | G ▷ C26×C3 ▷ C26 | |
| UCS | C1 ◁ C3 ◁ C32 | |
6046 conjugacy classes of subgroups.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
3^3,2^6|b^a=bc,d^a=de,e^a=d,f^a=fg,g^a=f,h^a=hi,i^a=h200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46180);