Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C3 ⋊ #13349 |
780 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 780 conjugacy classes of subgroups — too many to draw. Only its 81 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=h,c^2=h,d^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fgh,c^b=cf,d^b=dh,g^b=gh,d^c=dg,i^c=i^2,f^d=fh63 conjugacy classes — view the full character table.
| maximal subgroup of | #72248330, D3×#13349, #72248434 |
| maximal normal subgroup of | #72248330, D3×#13349, #72248434 |
| maximal quotient of | #72255264 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 88543);