Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C2×C4×H(3,𝔽3) ⋊ D4 |
| Sylow | H(3,𝔽3) ⋊ #394 |
621 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 621 conjugacy classes of subgroups — too many to draw. Only its 51 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|c^2=ef,b^a=bd,c^a=cf,g^a=g^2,h^a=g^2h,i^a=i^2,c^b=ce,g^b=g^2h^2,h^b=hi,i^b=i^2,g^d=g^2i^2,h^d=h^2i^2,h^g=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 12697);