Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | C32×H(5,𝔽2) ⋊ D3 |
390 conjugacy classes of subgroups; only the 36 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 36 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:
2^3,3^3,2^3|b^2=i,c^2=i,g^2=i,h^2=i,b^a=bc,c^a=ci,d^a=d^2,e^a=e^2,g^a=gh,h^a=hi,c^b=ci,e^d=ef,g^d=h,h^d=ghi,h^g=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72220369, #72220372, C2×#13147078 |
| maximal normal subgroup of | #72220372, C2×#13147078 |
| maximal quotient of | C2×#13147078 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 13141);