
Description


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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), 18. 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.

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These notations are used by the galois function.

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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



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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.

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

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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 $Y$.


$X\:Y$ 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 semidirect 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 $X\mathrm{wr}Y$.


$\frac{1}{m}\left[X\right]Y$ 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 $\mathrm{e1}\left(a\right)=\mathrm{e2}\left(b\right)$.

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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.



