Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| isoclinism | isoclinic class of #4626⋊C2 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4 · Schur multiplier (H2) = C23 |
| unambiguous semidirect product | (#560⋊C3)⋊C4 |
315 conjugacy classes of subgroups; only the 29 normal subgroups are shown.
This group has 315 conjugacy classes of subgroups — too many to draw. Only its 29 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:
2^4,3,2^4|a^2=d,f^2=h,g^2=i,b^a=bc,e^a=e^2,f^a=fgi,g^a=gi,h^a=hi,f^b=fi,g^b=ghi,f^c=fh,g^c=gi,f^e=gh,g^e=fg,h^e=i,i^e=hi32 conjugacy classes — view the full character table.
| maximal subgroup of | #72219746, C4×#4626⋊C2, #72243028, … |
| maximal normal subgroup of | #72219746, C4×#4626⋊C2 |
| maximal quotient of | #72219750, #72238101, #72238103, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1085907);