Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C42 · Schur multiplier (H2) = C22×C4 |
| unambiguous semidirect product | #10588329⋊C3 |
| Sylow | 3−1+2 ⋊ C4×C4⋊C4 |
620 conjugacy classes of subgroups.
| transitive degree | μt(G) = 576 |
Polycyclic presentation:
2^3,3,2^3,3^2|a^2=g,b^2=f,c^2=e,h^3=i,c^a=cg,h^a=h^2i^2,i^a=i^2,c^b=cg,h^d=hi^2200 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 7431);