Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | C33 ⋊ #473 |
622 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 622 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 96 |
Polycyclic presentation:
2^6,3^3|a^2=e,b^2=e,c^2=d,d^2=f,e^2=f,b^a=bd,c^a=cde,d^a=df,i^a=i^2,c^b=cde,d^b=df,g^b=g^2,e^c=ef,h^c=h^290 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 22328);