Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C3×C9 ⋊ #548 |
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D3×D9 |
| commutator, cosocle | C22×C3×C9 | C24 |
| Frattini | C22×C3 | C22×D32 |
| Fitting | C3×C42×C9 | C22 |
| socle | C22×C32 | C23×D3 |
| hypercenter | C42 | D3×D9 |
| hypocenter | C3×C9 | #548 |
| LCS | G ▷ C22×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C22 ◁ C42 | |
964 conjugacy classes of subgroups.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^6,3^3|a^2=f,c^2=f,d^2=ef,h^3=i,c^a=cf,g^a=g^2,c^b=cf,d^b=de,h^b=h^2i^2,i^b=i^2183 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 13553);