Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q16×H(3,𝔽3) ⋊ C4 |
| Sylow | H(3,𝔽3) ⋊ #444 |
280 conjugacy classes of subgroups; only the 47 normal subgroups are shown.
This group has 280 conjugacy classes of subgroups — too many to draw. Only its 47 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=d,b^2=f,c^2=f,d^2=f,e^2=f,c^a=cf,g^a=gh,h^a=gh^2i^2,c^b=ce,e^b=ef,e^c=ef,g^d=g^2,h^d=h^2i,h^g=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 13437);