Finite Group DatabasebetaFinite Group Databasebeta| order | 768 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C3 · Schur multiplier (H2) = C22 |
| unambiguous semidirect product | #32901⋊C3 |
| Fitting, Sylow | #32901 ⋊ C3 |
696 conjugacy classes of subgroups; only the 24 normal subgroups are shown.
This group has 696 conjugacy classes of subgroups — too many to draw. Only its 24 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=i,e^2=i,b^a=bfh,c^a=cdfg,d^a=ceg,f^a=fg,g^a=fh,c^b=cf,d^b=dg,e^b=eh,g^c=gi,f^d=fi43 conjugacy classes — view the full character table.
| maximal subgroup of | #72341550, #72341555, #72341563, … |
| maximal normal subgroup of | #72341550, #72341555, #72341563, … |
| maximal quotient of | #72341912 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(768, 1083987);