Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C3 |
No aliases for this group.
| Sylow | C33 ⋊ #442 |
1096 conjugacy classes of subgroups; only the 176 normal subgroups are shown.
This group has 1096 conjugacy classes of subgroups — too many to draw. Only its 176 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=e,b^2=e,c^2=d,d^2=f,c^a=cf,g^a=g^2,h^a=h^2,c^b=cd,d^b=df,i^b=i^2162 conjugacy classes — view the full character table.
| maximal subgroup of | C3⋊D3×#4711, D3×#10592568, #72245396 |
| maximal normal subgroup of | C3⋊D3×#4711, D3×#10592568, #72245396 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 25728);