Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C1 |
| unambiguous semidirect product | (PSU(3,𝔽2)⋊C3)⋊Q8 |
180 conjugacy classes of subgroups; only the 20 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 20 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 72 |
Polycyclic presentation:
2^3,3,2^3,3^2|a^2=c,b^2=c,e^2=g,f^2=g,b^a=bc,d^a=d^2,e^a=fg,f^a=eg,h^a=h^2i,e^d=ef,f^d=e,h^d=hi,f^e=fg,h^e=i^2,i^e=h,h^f=hi,i^f=hi^2,h^g=h^2,i^g=i^232 conjugacy classes — view the full character table.
| maximal subgroup of | #72220119, #72220125, Q8×AGL(2,𝔽3), … |
| maximal normal subgroup of | #72220119, #72220125, Q8×AGL(2,𝔽3), … |
| maximal quotient of | #72245017, #72268992 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 46190);