Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #10593631⋊C3 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22×C3 |
| unambiguous semidirect product | #10593631⋊C3 |
| Fitting | C32×#127 ⋊ D3 |
| derived | C3×C42⋊C3 ⋊ C22×C3 |
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C1 | G |
| commutator, cosocle, hypocenter | C3×C42⋊C3 | C22×C3 |
| Frattini, socle | C22×C3 | C2×C3×S4 |
| Fitting | C32×#127 | D3 |
| LCS | G ▷ C3×C42⋊C3 | |
| UCS | C1 | |
345 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 345 conjugacy classes of subgroups — too many to draw. Only its 25 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 36 |
Polycyclic presentation:
2^2,3^3,2^4|f^2=h,g^2=i,d^a=d^2,e^a=e^2,f^a=fg,g^a=gi,h^a=hi,f^b=fh,g^b=gi,d^c=de^2,f^d=ghi,g^d=fgh,h^d=i,i^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12897);