Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3×C4 · Schur multiplier (H2) = C23×C3×C4 |
| unambiguous semidirect product | #10592687⋊C3 |
| Fitting | C2×C4⋊C4×H(3,𝔽3) ⋊ C2 |
| Sylow | H(3,𝔽3) ⋊ #390 |
995 conjugacy classes of subgroups.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^3,3,2^3,3^2|b^2=g,c^2=e,b^a=bg,c^a=cf,h^a=h^2,i^a=i^2,c^b=cg,h^d=hi164 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 8689);