Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| Sylow | C33 ⋊ C2×AGL(1,ℤ/8) |
1090 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 1090 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
2^6,3^3|d^2=f,e^2=f,b^a=be,d^a=df,e^a=ef,g^a=g^2,h^a=h^2,i^a=h^2i,c^b=cf,e^b=ef,g^b=g^2,h^b=h^2i^2,h^e=h^2,i^e=i^278 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 47199);