Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C3 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | (#126⋊C3)⋊Q8 |
| unambiguous semidirect product | #46002⋊C3 |
| Fitting, Sylow | #46002 ⋊ C3 |
374 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 374 conjugacy classes of subgroups — too many to draw. Only its 60 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|b^2=gh,d^2=h,e^2=i,f^2=g,b^a=be,c^a=cgi,d^a=defi,e^a=ef,f^a=eg,g^a=i,h^a=ghi,i^a=gi,c^b=cg,d^b=dh,e^b=ei,f^b=fg,d^c=di,e^c=egi,f^c=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72258635 |
| maximal normal subgroup of | #72258635 |
| maximal quotient of | #72258643 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1084452);