(1) Arithmetic crystal class , line : In the reciprocalspace group of the arithmetic crystal class , the line has stabilizer and little cogroup . Therefore, the divisor is 12:6 = 2 and is running from 0 to .
The same result holds for the line in the reciprocalspace group of the arithmetic crystal class : the stabilizer generated by is of order 6, , the quotient is again , the parameter range is the same as for .
(2) Arithmetic crystal class , plane : In , the stabilizer of is generated by and the centring translation mod (integer translations). They generate a group of order 16; is of order 2. The fraction of the plane is of the area in the (centred) unit cell, as expressed by the parameter ranges . There are six arms of the star of : ; ; ; ; ; . Three of them (, and ) are represented in the boundaries of the representation domain: C = [Γ N P], B = [H N P] and J = [Γ H P], see Fig. 1.5.5.1. The areas of their parameter ranges are and , respectively; the sum is .
Arithmetic crystal class , the same result holds in the reciprocalspace group . The stabilizer generated by and by the centring translation mod (integer translations) forms a group of order 8; the order of the little cogroup . The quotient is again , the parameter range is the same as for but the plane belongs to the general position GP because the little cogroup is trivial.
(3) Arithmetic crystal class , reciprocalspace group , plane : the stabilizer of the plane A is generated by and , order 32, (sitesymmetry group) , order 2. Consequently, [Γ H N] is of the unit square : . In , the stabilizer of , here , is and , order 16, with the same group of order 2. Therefore, [Γ H_{2} H] is of the unit square in ; .
(4) Arithmetic crystal class , line : In the stabilizer is generated by and mod (integer translations), order 16, is of order 4. The divisor is 4 and thus . In the stabilizer is generated by and mod (integer translations), order 8, and , order 2; the divisor is 4 again and is restricted to the same range.
