Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| Fitting | C33×D4 ⋊ C23 |
| Sylow | C33 ⋊ C2×H(5,𝔽2) |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D33 |
| commutator, socle, cosocle | C2×C33 | C25 |
| Fitting | C33×D4 | C23 |
| hypercenter | D4 | D33 |
| hypocenter | C33 | C2×H(5,𝔽2) |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C2 ◁ D4 | |
4282 conjugacy classes of subgroups.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^6,3^3|e^a=ef,h^a=h^2,d^b=df,g^b=g^2,d^c=df,e^c=ef,i^c=i^2,e^d=ef120 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#10596675 |
| maximal normal subgroup of | D3×#10596675 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47350);