Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Sylow | C33 ⋊ #407 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2×C4 | D33 |
| commutator, socle | C22×C33 | C22×C4 |
| Fitting, cosocle | C2×C33×C4 | C23 |
| prime radical | C22×C3×D32 | C4 |
| hypocenter | C33 | #407 |
| LCS | G ▷ C22×C33 ▷ C33 | |
| UCS | C1 ◁ C2×C4 | |
950 conjugacy classes of subgroups.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^6,3^3|c^2=d,d^2=f,c^a=cf,i^a=i^2,c^b=ce,g^b=g^2,h^c=h^2156 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 21372);