Notes on the notation of the corepresentations in the Bilbao Crystallographic Server
The corepresentations of a magnetic group are obtained from the irreducible representations of its unitary subgroup. For the notation of the corepresentations of the little groups, the label of the corepresentation is obtained from the label (or labels) of the corresponding irreducible representation(s). In those cases (in some type III groups) in which the label of the kvector in the magnetic group is different from the label in the unitary subgroup, the notation includes as a prefix the label of the magnetic group.
In the tables of corepresentations and in the following examples, the BNS setting is assumed.
Examples of the notation used for the corepresentations of the little groups
Example 1: magnetic group Cmc2_{1}1' (N. 36.173)
It is a magnetic group of type II (gray group), and its unitary subgroup is the space group Cmc2_{1} (N. 36).

At point Γ:(0,0,0) in the reciprocal space of the magnetic group, the irreps are obtained from the irreps at the Γ point of the reciprocal space of the space group. All the irreps Γ_{1}, Γ_{2}, Γ_{3}, Γ_{4} and Γ_{5} are of type (a), and they induce the corepresentations with the same label.

At point T:(1,0,1/2) of the magnetic group, the corepresentations are obtained from the representations at point T:(1,0,1/2) of the space group.
The pair of irreps T_{1} and T_{3} are of type (c), and they induce the correpresentation T_{1}T_{3} in Cmc2_{1}1'.
The pair of irreps T_{2} and T_{4} are of type (c), and they induce the correpresentation T_{2}T_{4} in Cmc2_{1}1'.
The doublevalued irrep T_{5} is of type (b), and it induces the correpresentation T_{5}T_{5} in Cmc2_{1}1'.

At point H:(1,0,w) of the magnetic group, the corepresentations are obtained from the representations at point H:(1,0,w) of the space group.
All the irreps (H_{1}, H_{2}, H_{3}, H_{4} and H_{5}) are of type (x), i.e., there are no antiunitary operations in the magnetic group that transform H:(1,0,w) into (1,0,w) (mod translations of the reciprocal lattice). The labels of the corepresentations are (H_{1}, H_{2}, H_{3}, H_{4} and H_{5}).
Example 2: magnetic group P4'2_{1}'m (N. 113.269)
It is a magnetic group of type III whose unitary subgroup is the space group Cmm2 (N. 33). The unitary operations are not in the standard setting of the space group. The transformation matrix to the standard setting is (1/2a1/b,1/2a+1/2b,c;0,1/2,0)

At point F:(u,1/2,w) of the magnetic group, the corepresentations are obtained from the representations at point GP:(1/2+u,1/2u,w) of the space group in its standard setting. The two irreps GP_{1} and GP_{2} are of type (b). The induced corepresentations are denoted as (F)GP_{1}GP_{1} and (F)GP_{2}GP_{2}
Using this notation, on the one hand, the label gives information about the origin of the corepresentation (the labels of the irreps of the unitary subgroup) and, on the other hand, the prefix indicates the specific point in the reciprocal space of the magnetic group.

At point B:(0,v,w) of the magnetic group, the corepresentations are obtained from the representations at point GP:(v,v,w) of the space group. The two irreps GP_{1} and GP_{2} are of type (a). The induced corepresentations are denoted as (B)GP_{1} and (B)GP_{2}

At point GP:(u,v,w) of the magnetic group, the corepresentations are obtained from the representations at point GP:(u+v,u+v,w) of the space group. The two irreps GP_{1} and GP_{2} are of type (x). The induced corepresentations are denoted as GP_{1} and GP_{2}.
In this last case, as the labels in the magnetic group and in the unitary subgroup are the same, no prefix is added.
Note that, in this example, if the prefix is omited, the resulting labels of the corepresentations could be misleading.
Examples of the notation used for the full corepresentations
Example 3: magnetic group I4' (N. 79.27)
It is a magnetic group of type III whose unitary subgroup is the space group C2 (N. 5). The unitary operations are not in the standard setting of the space group. The transformation matrix to the standard setting is (ac,a,b;0,0,0)

In the white group I4, the point X in the reciprocal space has two branches in the star, X:(1/2,1/2,0),(1/2,1/2,0). These points correspond to the points M:(0,1,1/2) and A:(0,0,1/2), respectively, in the standard setting of its space group C2. Therefore, the corepresentations of the little group of X:(1/2,1/2,0) in the magnetic group are obtained from the irreps at M:(0,1,1/2) in the unitary subgroup and the corepresentations at X:(1/2,1/2,0) from the irreps at A:(0,0,1/2).
The corepresentations of the little group are denoted as (X)M_{1}, (X)M_{2}, (X)M_{3} and (X)A_{4} in the first case and as (X)A_{1}, (X)A_{2}, (X)A_{3} and (X)A_{4} in the second one.
The full representations at X are denoted as,
^{*}(X)M_{1}A_{1}, ^{*}(X)M_{2}A_{2}, ^{*}(X)M_{3}A_{4} and ^{*}(X)M_{4}A_{3}.
Note that the pairs of conjugated irreps that join to form the full corepresentations are (M_{1},A_{1}), (M_{2},A_{2}), (M_{3},A_{4}) and (M_{4},A_{3}).