Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | (C2×((C22⋊C9)⋊C2)⋊C3)⋊C4 |
| unambiguous semidirect product | ((C2×(C22⋊C9)⋊C2)⋊C4)⋊C3 |
554 conjugacy classes of subgroups; only the 58 normal subgroups are shown.
This group has 554 conjugacy classes of subgroups — too many to draw. Only its 58 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:
2^2,3,2^2,3^2,2^2|b^2=d,f^3=g,b^a=be,f^a=f^2g^2,g^a=g^2,h^a=hi,f^c=fg,h^f=i,i^f=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#63694⋊C3 |
| maximal normal subgroup of | C4×#63694⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12942);