Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | Q8×H(3,𝔽3) ⋊ D4 |
| Sylow | H(3,𝔽3) ⋊ #453 |
486 conjugacy classes of subgroups; only the 35 normal subgroups are shown.
This group has 486 conjugacy classes of subgroups — too many to draw. Only its 35 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^6,3^3|c^2=f,e^2=f,b^a=bd,e^a=ef,g^a=g^2,h^a=g^2h,i^a=i^2,c^b=ce,e^b=ef,g^b=g^2h^2,h^b=hi,i^b=i^2,d^c=df,e^c=ef,g^d=g^2i^2,h^d=h^2i^2,h^g=hi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12713);