Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2×C3 |
| unambiguous semidirect product | #10593663⋊C3 |
| Fitting | C32×H(5,𝔽2) ⋊ D3 |
340 conjugacy classes of subgroups; only the 33 normal subgroups are shown.
This group has 340 conjugacy classes of subgroups — too many to draw. Only its 33 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^2,3,2,3^2,2^3|b^2=i,d^2=i,g^2=i,h^2=i,b^a=bd,d^a=di,e^a=e^2,f^a=f^2,g^a=gh,h^a=hi,d^b=di,e^c=ef^2,g^e=h,h^e=ghi,h^g=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72220371, #72220372, C2×#10593663⋊C3 |
| maximal normal subgroup of | #72220372, C2×#10593663⋊C3 |
| maximal quotient of | C2×#10593663⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12994);