Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C2×C3×C42 |
No aliases for this group.
| Sylow | C33 ⋊ #376 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22×C4 | D3×C3⋊D3 |
| commutator | C2×C33 | C2×C42 |
| Frattini | C23 | C2×D3×C3⋊D3 |
| Fitting | C22×C33×C4 | C22 |
| socle, prime radical, cosocle | C23×C33 | C23 |
| hypocenter | C33 | #376 |
| LCS | G ▷ C2×C33 ▷ C33 | |
| UCS | C1 ◁ C22×C4 | |
1500 conjugacy classes of subgroups.
| transitive degree | μt(G) = 576 |
Polycyclic presentation:
2^6,3^3|a^2=f,b^2=d,c^2=e,b^a=be,g^a=g^2,h^a=h^2,i^b=i^2264 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 26314);