Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C2×C3×Q8×C9 ⋊ C22 |
| Sylow | C3×C9 ⋊ #585 |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C23×D3×D9 |
| commutator, cosocle | C2×C3×C9 | C25 |
| Frattini | C2×C3 | C23×D32 |
| Fitting | C2×C3×Q8×C9 | C22 |
| socle | C2×C32 | C24×D3 |
| hypercenter | C2×Q8 | D3×D9 |
| hypocenter | C3×C9 | #585 |
| LCS | G ▷ C2×C3×C9 ▷ C3×C9 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 | |
2576 conjugacy classes of subgroups.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^6,3^3|c^2=f,e^2=f,h^3=i,e^a=ef,g^a=g^2,d^b=df,h^b=h^2i^2,i^b=i^2,e^c=ef171 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30825);