Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | D4⋊#6188 |
| Sylow | H(3,𝔽3) ⋊ D4⋊C8 |
311 conjugacy classes of subgroups; only the 43 normal subgroups are shown.
This group has 311 conjugacy classes of subgroups — too many to draw. Only its 43 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 288 |
Polycyclic presentation:
2^6,3^3|a^2=c,c^2=f,d^2=e,b^a=bd,d^a=de,g^a=gh,h^a=gh^2i^2,d^b=de,g^c=g^2,h^c=h^2i,h^g=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 3010);