Finite Group DatabasebetaFinite Group Databasebeta| order | 1728 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | (D9⋊SL(2,𝔽3))⋊C4 |
| Fitting | C4×Q8×C9 ⋊ C2×C3 |
269 conjugacy classes of subgroups; only the 48 normal subgroups are shown.
This group has 269 conjugacy classes of subgroups — too many to draw. Only its 48 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:
2^2,3,2^3,3,2,3|b^2=d,e^2=h,f^2=h,g^3=i,b^a=bh,g^a=g^2i^2,i^a=i^2,e^c=ef,f^c=e,g^c=gi^2,f^e=fh86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(1728, 13172);