Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C32 · Schur multiplier (H2) = C22×C32×C42 |
| unambiguous semidirect product | C32⋊#5464 |
| Fitting | C32×H(3,𝔽4) ⋊ C3 |
| Sylow | H(3,𝔽4) ⋊ H(3,𝔽3) |
| subgroup | G/H | |
|---|---|---|
| center | C3 | C3×#5464 |
| commutator, cosocle | C3×H(3,𝔽4) | C32 |
| Frattini, socle | C22×C3 | C3×C22⋊A4 |
| Fitting | C32×H(3,𝔽4) | C3 |
| hypercenter | C32 | #5464 |
| hypocenter | H(3,𝔽4) | H(3,𝔽3) |
| LCS | G ▷ C3×H(3,𝔽4) ▷ H(3,𝔽4) | |
| UCS | C1 ◁ C3 ◁ C32 | |
404 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=i,i^a=hi,f^d=fh,g^d=gi,f^e=fi,g^e=ghi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 18329);