Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C3×C4 · Schur multiplier (H2) = C2×C4 |
| unambiguous semidirect product | #20608⋊C3 |
| Fitting, Sylow | #20608 ⋊ C3 |
718 conjugacy classes of subgroups; only the 32 normal subgroups are shown.
This group has 718 conjugacy classes of subgroups — too many to draw. Only its 32 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| transitive degree | μt(G) = 64 |
Polycyclic presentation:
3,2^8|b^2=h,e^2=i,b^a=bcdeh,c^a=cdefghi,d^a=cefhi,f^a=fg,g^a=fi,h^a=fghi,c^b=cf,d^b=dg,f^b=fi,g^c=gi,h^c=hi,f^d=fi64 conjugacy classes — view the full character table.
| maximal subgroup of | #72341274, #72341280, #72341287, … |
| maximal normal subgroup of | #72341274, #72341280, #72341287, … |
| maximal quotient of | #72341931 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083839);