Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| Sylow | C33 ⋊ #410 |
626 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 626 conjugacy classes of subgroups — too many to draw. Only its 60 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|b^2=e,c^2=f,b^a=bd,e^a=ef,h^a=h^2,i^a=h^2i,c^b=cf,d^b=df,h^b=h^2i^2,g^c=g^2,h^d=h^2,i^d=i^266 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 31521);