Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | 3−1+2×H(5,𝔽2) ⋊ C2 |
| Sylow | 3−1+2 ⋊ B(4,𝔽2) |
677 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 677 conjugacy classes of subgroups — too many to draw. Only its 102 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:
2^3,3,2^3,3^2|h^3=i,b^a=bf,c^a=ce,h^a=h^2i^2,i^a=i^2,e^b=eg,f^c=fg,h^d=hi^2116 conjugacy classes — view the full character table.
| maximal subgroup of | AGL(1,ℤ/9)×B(4,𝔽2), #72220135, #72245451, … |
| maximal normal subgroup of | AGL(1,ℤ/9)×B(4,𝔽2), C2×#13143663 |
| maximal quotient of | #72220135, C2×#13143663 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 9726);