Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C25 |
| unambiguous semidirect product | #10589757⋊C3 |
| Fitting | 3−1+2×#123 ⋊ C2 |
| Sylow | 3−1+2 ⋊ #547 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C23×AGL(1,ℤ/9) |
| commutator, cosocle | C22×C9 | C24×C3 |
| Frattini, socle | C22×C3 | C23×C3×D3 |
| Fitting | 3−1+2×#123 | C2 |
| hypercenter | #123 | AGL(1,ℤ/9) |
| hypocenter | C9 | C3×#547 |
| LCS | G ▷ C22×C9 ▷ C9 | |
| UCS | C1 ◁ C22 ◁ #123 | |
915 conjugacy classes of subgroups.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^4,3,2^2,3^2|c^2=g,d^2=g,h^3=i,b^a=bg,h^a=h^2i^2,i^a=i^2,d^b=dg,d^c=df,h^e=hi^2131 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 19129);