Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | #10588549⋊C3 |
| Fitting | 3−1+2×#106 ⋊ C2 |
| Sylow | 3−1+2 ⋊ #442 |
414 conjugacy classes of subgroups; only the 112 normal subgroups are shown.
This group has 414 conjugacy classes of subgroups — too many to draw. Only its 112 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^3,3,2^3,3^2|b^2=fg,c^2=e,f^2=g,h^3=i,b^a=bf,f^a=fg,h^a=h^2i^2,i^a=i^2,c^b=cf,f^c=fg,h^d=hi^2122 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,ℤ/9)×#442, #72244363, #72244374, … |
| maximal normal subgroup of | AGL(1,ℤ/9)×#442, #72244363, #72244374, … |
| maximal quotient of | #72244363, #72254708 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 8233);