Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C33 ⋊ #575 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C4 | D3×GO+(4,𝔽2) |
| commutator, cosocle | C3×C3⋊Dic3 | C24 |
| Frattini | C2 | C2×D3×GO+(4,𝔽2) |
| Fitting | C33×C4 | C2×D4 |
| socle | C2×C33 | C22×D4 |
| hypocenter | C33 | #575 |
| LCS | G ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C4 | |
946 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^6,3^3|d^2=f,e^2=f,b^a=be,e^a=ef,h^a=h^2,i^a=h^2i,c^b=cf,e^b=ef,h^b=h^2i^2,g^c=g^2,h^e=h^2,i^e=i^278 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46571);