Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C3 |
No aliases for this group.
| Fitting | C2×C33×Q8 ⋊ C22 |
| Sylow | C33 ⋊ C2×2−1+4 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C23×D3×C3⋊D3 |
| commutator, socle, cosocle | C2×C33 | C25 |
| Fitting | C2×C33×Q8 | C22 |
| hypercenter | C2×Q8 | D3×C3⋊D3 |
| hypocenter | C33 | C2×2−1+4 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C2×Q8 | |
5388 conjugacy classes of subgroups.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^6,3^3|c^2=f,d^2=f,e^a=ef,g^a=g^2,h^a=h^2,e^b=ef,i^b=i^2,d^c=df159 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47695);