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Transitive Groups Naming Scheme

Description

 • 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:

 A   : Alternating F   : Frobenius E   : Elementary S   : Symmetric AL  : Affine linear C   : Cyclic M   : Mathieu D   : Dihedral Q_8 : Quaternionic group

 • Except for dihedral and Frobenius group, a name of the form $X\left(n\right)$, where $X$ is a family name, denotes the $n$-th member of this family acting as a permutation group on $n$ points. For instance, $S\left(3\right)$ is the symmetric group on 3 elements. Moreover, ${X}_{n}$ denotes the same abstract group, but not necessarily with the same action. For instance, ${A}_{4}\left(6\right)$ is the alternating group on 4 elements acting transitively on a set of 6 elements. For dihedral and Frobenius groups, ${F}_{n}$ or ${\mathrm{D}}_{n}$ denotes the group of order $n$. For instance, $\mathrm{D}\left(4\right)={\mathrm{D}}_{8}\left(4\right)$ and ${\mathrm{D}}_{6}\left(6\right)$ 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 $X$ and $Y$ be groups. Then
 $\mathrm{XY}$ or $X.Y$ indicates a group with a normal subgroup of structure $X$, for which the corresponding quotient has structure $Y$.
 $X$ specifies that the  group is a split extension.
 ${X}_{x}Y$ denotes a direct product where the action is the natural action on the Cartesian product of the sets.
 $X[\frac{1}{m}]Y$ denotes a subdirect product corresponding to two epimorphisms $\mathrm{e1}$: $X\to F$ and $\mathrm{e2}$: $Y\to F$ where $F$ is a group of order $m$. In other words, the group consists of elements $a,b$ in the direct product ${X}_{x}Y$ such that $\mathrm{e1}\left(a\right)=\mathrm{e2}\left(b\right)$.
 ${X}^{n}$ is the direct product of $n$ groups of structure $X$.
 $X\mathrm{wr}Y$ denotes a wreath product.
 $[X]Y$ is an imprimitive group derived from a semi-direct product. The group $X$ is the intersection of the block stabilizers. See the paper by Conway, Hulpke, and McKay for more information. In particular $[{X}^{n}]Y$ (where $Y$ has degree $n$) is the permutational wreath product $X\mathrm{wr}Y$.
 $\frac{1}{{m}_{X}Y}$ denotes a subgroup of $[X]Y$. There exists two epimorphisms $\mathrm{e1}$: $X\to F$ and $\mathrm{e2}$: $Y\to F$ (where the order of $F$ is $m$, such that the group consists of elements $a,b$ in $[X]Y$ satisfying $\mathrm{e1}\left(a\right)=\mathrm{e2}\left(b\right)$.
 • 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.