Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C2 |
| unambiguous semidirect product | #33271⋊C3 |
| Fitting, Sylow | #33271 ⋊ C3 |
156 conjugacy classes of subgroups; only the 23 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 23 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 48 |
Polycyclic presentation:
3,2^8|c^2=fh,d^2=g,e^2=f,g^2=i,b^a=bf,c^a=egi,d^a=dhi,e^a=ceh,f^a=hi,h^a=fh,c^b=cf,d^b=dg,e^b=eh,g^b=gi,e^c=ei36 conjugacy classes — view the full character table.
| maximal subgroup of | #72340526, #72340598, #72340609 |
| maximal normal subgroup of | #72340526, #72340609 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084022);