This page briefly describes some of the notations introduced in the paper "On Transitive Permutation Groups" by J.H. Conway, A. Hulpke, and J. McKay, LMS J. Comput. Math. 1 (1998), 1-8. These notations are reminiscent of the notations used in "The Atlas of Finite Groups" by J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, and R.A. Wilson.

These notations are used by the galois function.

Capital letters denote families of groups:

Except for dihedral and Frobenius group, a name of the form , where  is a family name, denotes the -th member of this family acting as a permutation group on  points. For instance,  is the symmetric group on 3 elements. Moreover,  denotes the same abstract group, but not necessarily with the same action. For instance,  is the alternating group on 4 elements acting transitively on a set of 6 elements. For dihedral and Frobenius groups,  or  denotes the group of order . For instance,  and  is the dihedral group with six elements acting transitively on a set of 6 elements.

An integer n stands for a cyclic group with n elements.

Let  and  be groups. Then

 or  indicates a group with a normal subgroup of structure , for which the corresponding quotient has structure .

 specifies that the  group is a split extension.

  denotes a direct product where the action is the natural action on the Cartesian product of the sets.

  denotes a subdirect product corresponding to two epimorphisms :  and :  where  is a group of order . In other words, the group consists of elements  in the direct product  such that .

 is the direct product of  groups of structure .

  denotes a wreath product.

  is an imprimitive group derived from a semi-direct product. The group  is the intersection of the block stabilizers. See the paper by Conway, Hulpke, and McKay for more information. In particular  (where  has degree ) is the permutational wreath product .

  denotes a subgroup of . There exists two epimorphisms :  and :  (where the order of  is , such that the group consists of elements  in  satisfying .

Lower case letters are used to distinguish different groups arising from the same general construction. See the paper by Conway, Hulpke, and McKay for more information.