Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Sylow | H(3,𝔽3) ⋊ #327 |
391 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 391 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 144 |
Polycyclic presentation:
2^6,3^3|b^2=c,c^2=e,d^2=e,b^a=bd,c^a=cf,d^a=de,h^a=h^2,i^a=i^2,d^b=def,g^b=g^2,i^b=i^2,h^g=hi89 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 1215);