Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | C4⋊(#458⋊C3) |
| unambiguous semidirect product | #32927⋊C3 |
| Fitting, Sylow | #32927 ⋊ C3 |
322 conjugacy classes of subgroups; only the 26 normal subgroups are shown.
This group has 322 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
3,2^8|b^2=i,c^2=f,b^a=bghi,d^a=ef,e^a=dei,g^a=hi,h^a=ghi,c^b=cf,d^b=dg,e^b=eh,h^d=hi,g^e=gi43 conjugacy classes — view the full character table.
| maximal subgroup of | #72344089, #72344090, #72344098, … |
| maximal normal subgroup of | #72344089, #72344090, #72344098, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083989);