Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C3 · Schur multiplier (H2) = C24 |
| unambiguous semidirect product | 3−1+2⋊#564 |
| unambiguous semidirect product | (C9⋊#564)⋊C3 |
| Sylow | 3−1+2 ⋊ #564 |
544 conjugacy classes of subgroups; only the 202 normal subgroups are shown.
This group has 544 conjugacy classes of subgroups — too many to draw. Only its 202 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 576 |
Polycyclic presentation:
2^4,3,2^2,3^2|a^2=g,b^2=f,c^2=f,d^2=g,h^3=i,c^a=cg,d^a=df,h^a=h^2i^2,i^a=i^2,c^b=cfg,d^b=dg,h^e=hi^2113 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 19256);