Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #560⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C27 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #560⋊C27 |
| unambiguous semidirect product | (#126⋊C27)⋊C2 |
| derived | C42 ⋊ C22×C27 |
| Sylow | #560 ⋊ C27 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C9 | #560⋊C3 |
| commutator, hypocenter | C42 | C22×C27 |
| Frattini | C22×C9 | C22×A4 |
| Fitting | C9×#560 | C3 |
| socle | C22×C3 | C22×C22⋊C9 |
| prime radical | C3×#560 | C9 |
| cosocle | C42×C9 | C22×C3 |
| LCS | G ▷ C42 | |
| UCS | C1 ◁ C9 | |
171 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 171 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2^2,3^3,2^4|c^3=d,d^3=e,f^2=h,g^2=hi,f^a=fh,g^a=ghi,f^b=fhi,g^b=gi,f^c=fg,g^c=fhi,h^c=i,i^c=hi153 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2250);