Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22 · Schur multiplier (H2) = C22 |
No aliases for this group.
| Fitting | #2616 ⋊ D3 |
303 conjugacy classes of subgroups; only the 16 normal subgroups are shown.
This group has 303 conjugacy classes of subgroups — too many to draw. Only its 16 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:
2^3,3,2^5|b^2=i,c^2=i,e^2=i,f^2=i,b^a=bc,c^a=ci,d^a=d^2,e^a=ef,f^a=fi,g^a=gh,c^b=ci,e^b=eg,f^b=fh,e^d=f,f^d=efi,g^d=h,h^d=gh,f^e=fi27 conjugacy classes — view the full character table.
| maximal subgroup of | #72285397 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085626);