Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| isoclinism | isoclinic class of #6249⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | SU(3,𝔽2)⋊C8 |
| Sylow | H(3,𝔽3) ⋊ Q8⋊C8 |
127 conjugacy classes of subgroups; only the 27 normal subgroups are shown.
This group has 127 conjugacy classes of subgroups — too many to draw. Only its 27 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 216 |
Polycyclic presentation:
2^6,3^3|a^2=c,b^2=d,c^2=e,d^2=f,b^a=bcde,g^a=g^2h^2,h^a=g^2,i^a=i^2,c^b=cef,g^b=g^2,h^b=ghi^2,i^b=i^2,g^c=g^2hi^2,h^c=ghi,g^e=g^2i^2,h^e=h^2i^2,h^g=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12238);