Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| Fitting | C2×C3×#354 ⋊ C2 |
| Sylow | C3 ⋊ #13050 |
732 conjugacy classes of subgroups; only the 95 normal subgroups are shown.
This group has 732 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 192 |
Polycyclic presentation:
2^8,3|a^2=f,b^2=df,c^2=h,e^2=h,b^a=bd,c^a=ce,e^a=egh,g^a=gh,c^b=ch,e^b=egh,i^b=i^2,d^c=dg,e^c=eh,f^c=fg,e^d=eh,f^e=fh60 conjugacy classes — view the full character table.
| maximal subgroup of | #72248492, #72248499, D3×#13050 |
| maximal normal subgroup of | #72248492, #72248499, D3×#13050 |
| maximal quotient of | #72270236 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 86883);