Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #32879⋊C3 |
| Fitting, Sylow | #32879 ⋊ C3 |
716 conjugacy classes of subgroups; only the 25 normal subgroups are shown.
This group has 716 conjugacy classes of subgroups — too many to draw. Only its 25 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=i,f^2=i,b^a=bgh,c^a=cdg,d^a=degi,e^a=dghi,f^a=fg,g^a=gh,h^a=g,c^b=cf,d^b=dg,e^b=eh,f^b=fi,f^c=fi38 conjugacy classes — view the full character table.
| maximal subgroup of | #72241465, SD16×#120⋊C3, #72241494, … |
| maximal normal subgroup of | SD16×#120⋊C3, #72241494, #72340611 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083973);