Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C32 · Schur multiplier (H2) = C2×C32 |
| unambiguous semidirect product | #458⋊H(3,𝔽3) |
| unambiguous semidirect product | (C3×#458⋊C3)⋊C3 |
| Fitting | C32×#458 ⋊ C3 |
| Sylow | #458 ⋊ H(3,𝔽3) |
259 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 259 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2,3^3,2^5|a^2=i,e^a=eg,f^a=fgh,c^b=cd,e^b=ef,f^b=e,g^b=hi,h^b=ghi,h^e=hi,g^f=gi,h^f=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 7324);