Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | Q16×H(3,𝔽3) ⋊ C22 |
| Sylow | H(3,𝔽3) ⋊ C2×SD32 |
375 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 375 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=f,d^2=ef,e^2=f,c^a=cd,d^a=de,e^a=ef,g^a=g^2,i^a=i^2,c^b=cd,d^b=de,e^b=ef,h^b=h^2,i^b=i^2,d^c=de,e^c=ef,h^g=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 5340);