Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #32896⋊C3 |
| Fitting, Sylow | #32896 ⋊ C3 |
670 conjugacy classes of subgroups; only the 22 normal subgroups are shown.
This group has 670 conjugacy classes of subgroups — too many to draw. Only its 22 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
Polycyclic presentation:
3,2^8|b^2=i,f^2=i,g^2=i,b^a=bfgi,c^a=dgi,d^a=ce,e^a=cdfgi,f^a=g,g^a=fhi,h^a=fg,c^b=cf,d^b=dg,e^b=eh,f^b=fi,g^b=gi,f^c=fi,h^c=hi,g^d=gi,f^e=fi34 conjugacy classes — view the full character table.
| maximal subgroup of | #72233783, #72236802, #72236874, … |
| maximal normal subgroup of | #72233783, #72236802, #72236874, … |
| maximal quotient of | #72341821, #72341955, C2×#32896⋊C3 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083982);