In two dimensions, the hyperoctahedral group structure is the abstract dihedral group of order eight, describing the symmetry of a square, or 2-orthoplex.
32.
Nor need a " p "-group be abelian; the dihedral group Dih 4 of order 8 is a non-abelian 2-group.
33.
Its full automorphism group is isomorphic to the dihedral group D 8 of order 16, the group of symmetries of an octagon, including both rotations and reflections.
34.
If the automorphisms form a nontrivial abelian group, the distinguishing number is two, and if it forms a dihedral group then the distinguishing number is at most three.
35.
The trick is just going to be when we put the three copies of the field together, do we do it like the dihedral group or like the quaternion group.
36.
It seems not, because both the cyclic group of order 10 and the dihedral group of order 10 could then be written as C _ 5 : C _ 2.
37.
It follows also that any group which is virtually cyclic ( contains a copy of \ mathbb Z of finite index ) is also hyperbolic, for example the infinite dihedral group.
38.
The character table does not in general determine the group up to isomorphism : for example, the quaternion group and the dihedral group of elements,, have the same character table.
39.
:The dihedral group of a regular " n " sided polygon is generated by the reflections in the perpendicular bisectors of its sides and the bisectors of its interior angles.
40.
This group is isomorphic to the dihedral group of order 6, the group of reflection and rotation symmetries of an equilateral triangle, since these symmetries permute the three vertices of the triangle.
