Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C32 |
| unambiguous semidirect product | #401⋊H(3,𝔽3) |
| unambiguous semidirect product | (C3×#401⋊C3)⋊C3 |
| Fitting | C32×#401 ⋊ C3 |
| Sylow | #401 ⋊ H(3,𝔽3) |
190 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 190 conjugacy classes of subgroups — too many to draw. Only its 29 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:
2,3^3,2^5|a^2=gh,e^2=g,f^2=gi,e^a=ehi,f^a=fi,c^b=cd,e^b=ef,f^b=ehi,g^b=ghi,h^b=i,i^b=hi,f^e=fgh98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 7315);