Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C3 |
| unambiguous semidirect product | #10589409⋊C3 |
| Sylow | H(3,𝔽3) ⋊ #330 |
266 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 266 conjugacy classes of subgroups — too many to draw. Only its 66 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:
2^2,3,2^4,3^2|a^2=e,b^2=g,d^2=f,e^2=g,b^a=bd,d^a=dg,h^a=h^2,i^a=i^2,d^b=df,e^b=efg,h^c=hi113 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 1685);