Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| Sylow | H(3,𝔽3) ⋊ #387 |
431 conjugacy classes of subgroups; only the 57 normal subgroups are shown.
This group has 431 conjugacy classes of subgroups — too many to draw. Only its 57 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=f,b^2=f,c^2=e,b^a=bd,g^a=g^2,h^a=g^2h,i^a=i^2,c^b=cf,g^b=g^2h^2,h^b=hi,i^b=i^2,g^d=g^2i^2,h^d=h^2i^2,h^g=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12675);