Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C3×C4 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #12774⋊C3 |
| Fitting, Sylow | #12774 ⋊ C3 |
| derived | H(3,𝔽4) ⋊ C3×C4 |
237 conjugacy classes of subgroups; only the 17 normal subgroups are shown.
This group has 237 conjugacy classes of subgroups — too many to draw. Only its 17 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 32 |
Polycyclic presentation:
3,2^8|b^2=g,e^2=i,f^2=i,g^2=i,b^a=bcfgi,c^a=dhi,d^a=cdfgi,e^a=fh,f^a=ef,g^a=eg,c^b=ce,d^b=df,e^b=eh,h^b=hi,e^c=ei,f^c=fhi,g^c=ghi,e^d=eh,f^d=fi,g^d=gi,f^e=fi,g^e=gi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72268241 |
| maximal normal subgroup of | #72268241 |
| maximal quotient of | #72227440, #72268602 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083566);