Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Fitting | C3×#2238 ⋊ C2 |
| Sylow | C3 ⋊ #33688 |
564 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 564 conjugacy classes of subgroups — too many to draw. Only its 121 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:
2^8,3|d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,f^b=fh,d^c=dg,e^c=eh,f^c=fg,g^c=gh,i^c=i^2,f^d=fh,i^d=i^254 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#33688, #72219793, #72270555, … |
| maximal normal subgroup of | D3×#33688, #72270555, #72270600, … |
| maximal quotient of | #72219793, #72258722 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 324797);