Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| Fitting | C3×#3549 ⋊ C2 |
| Sylow | C3 ⋊ #33570 |
1212 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 1212 conjugacy classes of subgroups — too many to draw. Only its 167 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=h,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,i^a=i^2,d^b=dh,e^b=eh,f^b=fgh,e^c=eg,f^c=fh,f^e=fh72 conjugacy classes — view the full character table.
| maximal subgroup of | D3×#33570, #72246069, #72246087, … |
| maximal normal subgroup of | D3×#33570, #72246069, #72246087, … |
| maximal quotient of | #72254244, #72255501 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 323769);