Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C33 ⋊ C2×AGL(1,ℤ/8) |
| subgroup | G/H | |
|---|---|---|
| center | C2 | C2×#88955 |
| commutator, cosocle | C33×C4 | C24 |
| Frattini | C4 | C2×D33 |
| Fitting | C2×C33×C4 | C23 |
| socle | C2×C33 | C22×D4 |
| hypercenter | C2×C4 | D33 |
| hypocenter | C33 | C2×AGL(1,ℤ/8) |
| LCS | G ▷ C33×C4 ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ C2×C4 | |
1618 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^6,3^3|b^2=e,c^2=e,e^2=f,b^a=be,c^a=ce,e^a=ef,h^a=h^2,d^b=df,g^b=g^2,d^c=df,i^c=i^2102 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#10595100, #72220384, #72220393 |
| maximal normal subgroup of | D3×#10595100, #72220384, #72220393 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 36651);