Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| Fitting | C22×H(3,𝔽3) ⋊ SD16 |
| Sylow | H(3,𝔽3) ⋊ #450 |
416 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 416 conjugacy classes of subgroups — too many to draw. Only its 37 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:
2^6,3^3|a^2=f,d^2=f,b^a=bd,c^a=ce,d^a=df,g^a=gh,h^a=gh^2i,d^b=df,g^b=g^2,h^b=hi^2,i^b=i^2,g^d=hi^2,h^d=g^2,g^f=g^2i^2,h^f=h^2i,h^g=hi41 conjugacy classes — view the full character table.
| automorphism group of | #88588, #88589, #88590, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 30160);