Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| Fitting | Q16×H(3,𝔽3) ⋊ C4 |
| Sylow | H(3,𝔽3) ⋊ #360 |
211 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 211 conjugacy classes of subgroups — too many to draw. Only its 25 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=c,b^2=f,d^2=e,e^2=f,b^a=bd,d^a=de,e^a=ef,g^a=gh,h^a=gh^2i^2,c^b=cf,d^b=def,e^b=ef,g^c=g^2,h^c=h^2i,h^g=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 2993);