# Permutation Representations of Linear Groups

Each of the functions in this family returns two values:

**(a)**
A permutation group $G$ corresponding to the action of a designated matrix group $M$ on a vector space $V$; and

**(b)**
An indexed set of affine or projective points on which $M$ acts, such that the indexing gives the correspondence between this set and the $G$-set of $M$.

Furthermore, most of the function in this family are parameterized by two objects: the *degree* and the *coefficient field* of the matrix group. These can be supplied in one of the following three forms:

**(i)**
Integers $n$ and $q$ corresponding to the degree and the field ${\bf F}_{q}$ of $M$ (${\bf F}_{q^2}$ in the case of the unitary groups).

**(ii)**
An integer $n$ and a finite field $K$ corresponding to the degree and the coefficient field of $M$.

**(iii)**
A vector space $V = K^n$ on which $M$ naturally acts.

The Suzuki group, however, is only parametrised by the field, as the degree is always four. As such, it can be described by the integer $q$, the field $K = {\bf F}_{q}$, or the vector space $K^4$.

## `AffineGeneralLinearGroup(arguments)`

## `AGL(arguments)`

## `AffineGeneralLinearGroup(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `AffineGeneralLinearGroup(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `AffineGeneralLinearGroup(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

## `AGL(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `AGL(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `AGL(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

Construct the affine general linear group $G = {\operatorname{AGL}}(n, q)$, i.e., the group corresponding to the action of ${\operatorname{GL}}(n, q)$ on the affine points of the $n$-dimensional vector space $V$ over $K = {\bf F}_{q}$. The function returns:

**(a)**
The group $G$;

**(b)**
An indexed set giving the correspondence between the affine points and the $G$-set of $G$.

## `AffineSpecialLinearGroup(arguments)`

## `ASL(arguments)`

## `AffineSpecialLinearGroup(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `AffineSpecialLinearGroup(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `AffineSpecialLinearGroup(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

## `ASL(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `ASL(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `ASL(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

Construct the affine special linear group $G = {\operatorname{ASL}}(n, q)$, i.e., the group corresponding to the action of ${\operatorname{SL}}(n, q)$ on the affine points of the $n$-dimensional vector space $V$ over $K = {\bf F}_{q}$. The function returns:

**(a)**
The group $G$;

**(b)**
An indexed set giving the correspondence between the affine points and the $G$-set of $G$.

## `AffineGammaLinearGroup(arguments)`

## `AGammaL(arguments)`

## `AffineGammaLinearGroup(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `AffineGammaLinearGroup(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `AffineGammaLinearGroup(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

## `AGammaL(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `AGammaL(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `AGammaL(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

Construct the affine gamma linear group $G = {\operatorname{A\Gamma L}}(n, q)$, i.e., the group corresponding to the action of ${\operatorname{\Gamma L}}(n, q)$ (the automorphism group of ${\operatorname{GL}}(n, q)$) on the affine points of the $n$-dimensional vector space $V$ over $K = {\bf F}_{q}$. The function returns:

**(a)**
The group $G$;

**(b)**
An indexed set giving the correspondence between the points and the $G$-set of $G$.

## `AffineSigmaLinearGroup(arguments)`

## `ASigmaL(arguments)`

## `AffineSigmaLinearGroup(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `AffineSigmaLinearGroup(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `AffineSigmaLinearGroup(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

## `ASigmaL(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `ASigmaL(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `ASigmaL(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

Construct the affine sigma linear group $G = {\operatorname{A\Sigma L}}(n, q)$, i.e., the group corresponding to the action of ${\operatorname{\Sigma L}}(n, q)$ (the automorphism group of ${\operatorname{SL}}(n, q)$) on the affine points of the $n$-dimensional vector space $V$ over $K = {\bf F}_{q}$. The function returns:

**(a)**
The group $G$;

**(b)**
An indexed set giving the correspondence between the points and the $G$-set of $G$.

## `AffineSymplecticGroup(arguments)`

## `ASp(arguments)`

## `AffineSymplecticGroup(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `AffineSymplecticGroup(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `AffineSymplecticGroup(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

## `ASp(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `ASp(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `ASp(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

Construct the affine symplectic linear group $G$, i.e., the group corresponding to the action of $Sp(n, q)$ on the affine points of the $n$-dimensional vector space $V$ over $K = {\bf F}_{q}$. The function returns:

**(a)**
The group $G$;

**(b)**
An indexed set giving the correspondence between the affine points and the $G$-set of $G$.

## `AffineSigmaSymplecticGroup(arguments)`

## `ASigmaSp(arguments)`

## `AffineSigmaSymplecticGroup(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `AffineSigmaSymplecticGroup(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `AffineSigmaSymplecticGroup(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

## `ASigmaSp(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `ASigmaSp(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `ASigmaSp(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

Construct the affine sigma symplectic linear group $G$, i.e., the group corresponding to the action of $Sp(n, q)$ on the affine points of the $n$-dimensional vector space $V$ over $K = {\bf F}_{q}$, plus the action of a field automorphism. The function returns:

**(a)**
The group $G$;

**(b)**
An indexed set giving the correspondence between the affine points and the $G$-set of $G$.

## `ProjectiveGeneralLinearGroup(arguments)`

## `PGL(arguments)`

## `ProjectiveGeneralLinearGroup(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveGeneralLinearGroup(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveGeneralLinearGroup(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

## `PGL(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `PGL(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `PGL(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

Construct the projective general linear group $G = {\operatorname{PGL}}(n, q)$, i.e., the group corresponding to the action of ${\operatorname{GL}}(n, q)$ on the projective points of the $n$-dimensional vector space $V$ over $K = {\bf F}_{q}$, where $n \geq 2$ and $q$ is a prime power. The function returns:

**(a)**
The group $G$;

**(b)**
An indexed set of the generators of the 1-dimensional subspaces of $K^{(n)}$, giving the correspondence between these vectors and the $G$-set of $G$.

## `ProjectiveSpecialLinearGroup(arguments)`

## `PSL(arguments)`

## `ProjectiveSpecialLinearGroup(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveSpecialLinearGroup(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveSpecialLinearGroup(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

## `PSL(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `PSL(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `PSL(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

Construct the projective special linear group $G = {\operatorname{PSL}}(n, q)$, i.e., the group corresponding to the action of ${\operatorname{SL}}(n, q)$ on the projective points of the $n$-dimensional vector space $V$ over $K = {\bf F}_{q}$, where $n \geq 2$ and $q$ is a prime power. The function returns:

**(a)**
The group $G$;

**(b)**
An indexed set of the generators of the 1-dimensional subspaces of $K^{(n)}$, giving the correspondence between these vectors and the $G$-set of $G$.

## `ProjectiveGammaLinearGroup(arguments)`

## `PGammaL(arguments)`

## `ProjectiveGammaLinearGroup(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveGammaLinearGroup(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveGammaLinearGroup(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

## `PGammaL(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `PGammaL(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `PGammaL(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

Construct an automorphism group $G = {\operatorname{P\Gamma L}}(n, q)$ of the projective general linear group $B = {\operatorname{PGL}}(n, q)$, by adding the field automorphisms of ${\bf F}_{q}$ to $B$. The permutation action corresponds to the natural action on 1-dimensional subspaces of the $n$-dimensional vector space $V$ over the field $K = {\bf F}_{q}$, where $n \geq 2$ and $q$ is a prime power. The function returns:

**(a)**
The group $G$;

**(b)**
An indexed set giving the correspondence between the points and the $G$-set of $G$.

## `ProjectiveSigmaLinearGroup(arguments)`

## `PSigmaL(arguments)`

## `ProjectiveSigmaLinearGroup(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveSigmaLinearGroup(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveSigmaLinearGroup(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

## `PSigmaL(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `PSigmaL(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `PSigmaL(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

Construct an automorphism group $G = {\operatorname{P\Sigma L}}(n, q)$ of the projective special linear group $B = {\operatorname{PSL}}(n, q)$, by adding the field automorphisms of ${\bf F}_{q}$ to $B$. The permutation action corresponds to the natural action on 1-dimensional subspaces of the $n$-dimensional vector space $V$ over the field $K = {\bf F}_{q}$, where $n \geq 2$ and $q$ is a prime power. The function returns:

**(a)**
The group $G$;

**(b)**
An indexed set giving the correspondence between the points and the $G$-set of $G$.

## `ProjectiveGeneralUnitaryGroup(arguments)`

## `PGU(arguments)`

## `ProjectiveGeneralUnitaryGroup(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveGeneralUnitaryGroup(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveGeneralUnitaryGroup(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

## `PGU(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `PGU(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `PGU(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

Construct the projective general unitary group $G = {\operatorname{PGU}}(n, q)$ corresponding to the $n$-dimensional vector space $V$ over the field $K = {\bf F}_{q^2}$, where $n \geq 2$ and $q$ is a prime power. The function returns:

**(a)**
The group $G$;

**(b)**
An indexed set of the generators of the 1-dimensional subspaces of $K^{(n)}$, giving the correspondence between these vectors and the $G$-set of $G$.

## `ProjectiveSpecialUnitaryGroup(arguments)`

## `PSU(arguments)`

## `ProjectiveSpecialUnitaryGroup(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveSpecialUnitaryGroup(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveSpecialUnitaryGroup(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

## `PSU(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `PSU(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `PSU(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

Construct the projective special unitary group $G = {\operatorname{PSU}}(n, q)$ corresponding to the $n$-dimensional vector space $V$ over the field $K = {\bf F}_{q^2}$, where $n \geq 2$ and $q$ is a prime power. The function returns:

**(a)**
The group $G$;

**(b)**
An indexed set of the generators of the 1-dimensional subspaces of $V$, giving the correspondence between these vectors and the $G$-set of $G$.

## `ProjectiveGammaUnitaryGroup(arguments)`

## `PGammaU(arguments)`

## `ProjectiveGammaUnitaryGroup(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveGammaUnitaryGroup(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveGammaUnitaryGroup(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

## `PGammaU(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `PGammaU(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `PGammaU(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

Construct an automorphism group $G = {\operatorname{P\Gamma U}}(n, q)$ of the projective general unitary group $B = {\operatorname{PGU}}(n, q)$, by adding the field automorphisms of ${\bf F}_{q^2}$ to $B$. The permutation action corresponds to the natural action on 1-dimensional subspaces of the $n$-dimensional vector space $V$ over the field $K = {\bf F}_{q^2}$, where $n \geq 2$ and $q$ is a prime power. The function returns:

**(a)**
The group $G$;

**(b)**
An indexed set giving the correspondence between the points and the $G$-set of $G$.

## `ProjectiveSigmaUnitaryGroup(arguments)`

## `PSigmaU(arguments)`

## `ProjectiveSigmaUnitaryGroup(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveSigmaUnitaryGroup(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveSigmaUnitaryGroup(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

## `PSigmaU(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `PSigmaU(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `PSigmaU(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

Construct the automorphism group $G = {\operatorname{P\Sigma U}}(n, q)$ of the projective special unitary group $B = {\operatorname{PSU}}(n, q)$, by adding the field automorphisms of ${\bf F}_{q^2}$ to $B$. The permutation action corresponds to the natural action on 1-dimensional subspaces of the $n$-dimensional vector space $V$ over the field $K = {\bf F}_{q^2}$, where $n \geq 2$ and $q$ is a prime power. The function returns:

**(a)**
The group $G$;

**(b)**
An indexed set giving the correspondence between the points and the $G$-set of $G$.

## `ProjectiveSymplecticGroup(arguments)`

## `PSp(arguments)`

## `ProjectiveSymplecticGroup(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveSymplecticGroup(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveSymplecticGroup(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

## `PSp(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `PSp(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `PSp(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

Construct the projective symplectic group $G = {\operatorname{PSp}}(n, q)$, where $K = {\bf F}_{q}$, $V$ is an $n$-dimensional vector space over $K$, and $n$ is an even integer greater than or equal to 4. The function returns:

**(a)**
The group $G$;

**(b)**
An indexed set of the generators of the 1-dimensional subspaces of $K^{(n)}$, giving the correspondence between these vectors and the $G$-set of $G$.

## `ProjectiveSigmaSymplecticGroup(arguments)`

## `PSigmaSp(arguments)`

## `ProjectiveSigmaSymplecticGroup(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveSigmaSymplecticGroup(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveSigmaSymplecticGroup(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

## `PSigmaSp(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `PSigmaSp(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `PSigmaSp(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

Construct the group $G = {\operatorname{P\Sigma Sp}}(n, q)$ of the projective symplectic group ${\operatorname{PSp}}(n, q)$ extended by field automorphisms of $K = {\bf F}_{q}$, where $V$ is an $n$-dimensional vector space over $K$, and $n$ is an even integer greater than or equal to 4. The function returns:

**(a)**
The group $G$;

**(b)**
An indexed set giving the correspondence between the points and the $G$-set of $G$.

## `ProjectiveGeneralOrthogonalGroup(arguments)`

## `PGO(arguments)`

## `ProjectiveGeneralOrthogonalGroup(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveGeneralOrthogonalGroup(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveGeneralOrthogonalGroup(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

## `PGO(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `PGO(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `PGO(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

Construct the projective general orthogonal group $G = {\operatorname{PGO}}(n, q)$, where $K = {\bf F}_{q}$, $V$ is an $n$-dimensional vector space over $K$, and $n$ is an odd integer greater than or equal to 3. The function returns:

**(a)**
The group $G$;

**(b)**
An indexed set of the generators of the 1-dimensional subspaces of $K^{(n)}$, giving the correspondence between these vectors and the $G$-set of $G$.

## `ProjectiveGeneralOrthogonalGroupPlus(arguments)`

## `PGOPlus(arguments)`

## `ProjectiveGeneralOrthogonalGroupPlus(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveGeneralOrthogonalGroupPlus(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveGeneralOrthogonalGroupPlus(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

## `PGOPlus(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `PGOPlus(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `PGOPlus(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

Construct the projective general orthogonal group $G = {\operatorname{PGO}}^+(n, q)$, where $K = {\bf F}_{q}$, $V$ is an $n$-dimensional vector space over $K$, and $n$ is an even integer greater than or equal to 2. The function returns:

**(a)**
The group $G$;

**(b)**
An indexed set of the generators of the 1-dimensional subspaces of $K^{(n)}$, giving the correspondence between these vectors and the $G$-set of $G$.

## `ProjectiveGeneralOrthogonalGroupMinus(arguments)`

## `PGOMinus(arguments)`

## `ProjectiveGeneralOrthogonalGroupMinus(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveGeneralOrthogonalGroupMinus(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveGeneralOrthogonalGroupMinus(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

## `PGOMinus(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `PGOMinus(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `PGOMinus(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

Construct the projective general orthogonal group $G = {\operatorname{PGO}}^-(n, q)$, where $K = {\bf F}_{q}$, $V$ is an $n$-dimensional vector space over $K$, and $n$ is an even integer greater than or equal to 2. The function returns:

**(a)**
The group $G$;

**(b)**
An indexed set of the generators of the 1-dimensional subspaces of $K^{(n)}$, giving the correspondence between these vectors and the $G$-set of $G$.

## `ProjectiveSpecialOrthogonalGroup(arguments)`

## `PSO(arguments)`

## `ProjectiveSpecialOrthogonalGroup(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveSpecialOrthogonalGroup(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveSpecialOrthogonalGroup(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

## `PSO(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `PSO(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `PSO(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

Construct the projective special orthogonal group $G = {\operatorname{PSO}}(n, q)$, where $K = {\bf F}_{q}$, $V$ is an $n$-dimensional vector space over $K$, and $n$ is an odd integer greater than or equal to 3. The function returns:

**(a)**
The group $G$;

**(b)**
An indexed set of the generators of the 1-dimensional subspaces of $K^{(n)}$, giving the correspondence between these vectors and the $G$-set of $G$.

## `ProjectiveSpecialOrthogonalGroupPlus(arguments)`

## `PSOPlus(arguments)`

## `ProjectiveSpecialOrthogonalGroupPlus(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveSpecialOrthogonalGroupPlus(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveSpecialOrthogonalGroupPlus(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

## `PSOPlus(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `PSOPlus(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `PSOPlus(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

Construct the projective special orthogonal group $G = {\operatorname{PSO}}^+(n, q)$, where $K = {\bf F}_{q}$, $V$ is an $n$-dimensional vector space over $K$, and $n$ is an even integer greater than or equal to 2. The function returns:

**(a)**
The group $G$;

**(b)**
An indexed set of the generators of the 1-dimensional subspaces of $K^{(n)}$, giving the correspondence between these vectors and the $G$-set of $G$.

## `ProjectiveSpecialOrthogonalGroupMinus(arguments)`

## `PSOMinus(arguments)`

## `ProjectiveSpecialOrthogonalGroupMinus(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveSpecialOrthogonalGroupMinus(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveSpecialOrthogonalGroupMinus(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

## `PSOMinus(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `PSOMinus(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `PSOMinus(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

Construct the projective general orthogonal group $G = {\operatorname{PSO}}^-(n, q)$, where $K = {\bf F}_{q}$, $V$ is an $n$-dimensional vector space over $K$, and $n$ is an even integer greater than or equal to 2. The function returns:

**(a)**
The group $G$;

**(b)**
An indexed set of the generators of the 1-dimensional subspaces of $K^{(n)}$, giving the correspondence between these vectors and the $G$-set of $G$.

## `ProjectiveOmega(arguments)`

## `POmega(arguments)`

## `ProjectiveOmega(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveOmega(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveOmega(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

## `POmega(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `POmega(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `POmega(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

Construct the projective orthogonal group $G = {\operatorname{P}\Omega}(n, q)$, where $K = {\bf F}_{q}$, $V$ is an $n$-dimensional vector space over $K$, and $n$ is an odd integer greater than or equal to 3. The function returns:

**(a)**
The group $G$;

**(b)**
An indexed set of the generators of the 1-dimensional subspaces of $K^{(n)}$, giving the correspondence between these vectors and the $G$-set of $G$.

## `ProjectiveOmegaPlus(arguments)`

## `POmegaPlus(arguments)`

## `ProjectiveOmegaPlus(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveOmegaPlus(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveOmegaPlus(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

## `POmegaPlus(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `POmegaPlus(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `POmegaPlus(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

Construct the projective orthogonal group $G = {\operatorname{P}\Omega}(n, q)$, where $K = {\bf F}_{q}$, $V$ is an $n$-dimensional vector space over $K$, and $n$ is an even integer greater than or equal to 2. The function returns:

**(a)**
The group $G$;

**(b)**
An indexed set of the generators of the 1-dimensional subspaces of $K^{(n)}$, giving the correspondence between these vectors and the $G$-set of $G$.

## `ProjectiveOmegaMinus(arguments)`

## `POmegaMinus(arguments)`

## `ProjectiveOmegaMinus(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveOmegaMinus(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveOmegaMinus(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

## `POmegaMinus(n, q): RngIntElt, RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `POmegaMinus(n, K): RngIntElt, FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `POmegaMinus(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

Construct the projective orthogonal group $G = {\operatorname{P}\Omega}(n, q)$, where $K = {\bf F}_{q}$, $V$ is an $n$-dimensional vector space over $K$, and $n$ is an even integer greater than or equal to 2. The function returns:

**(a)**
The group $G$;

**(b)**
An indexed set of the generators of the 1-dimensional subspaces of $K^{(n)}$, giving the correspondence between these vectors and the $G$-set of $G$.

## `ProjectiveSuzukiGroup(arguments)`

## `PSz(arguments)`

## `ProjectiveSuzukiGroup(q): RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveSuzukiGroup(K): FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `ProjectiveSuzukiGroup(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

## `PSz(q): RngIntElt -> GrpPerm, {@ ModTupFldElt  @}`

## `PSz(K): FldFin -> GrpPerm, {@ ModTupFldElt  @}`

## `PSz(V): ModTupRng -> GrpPerm, {@ ModTupFldElt  @}`

Construct the permutation representation $G = {\operatorname{PSz}}(q)$ of the Suzuki simple group ${\operatorname{Sz}}(q)$, given by its action on projective points, where $q$ is of the form $2^{2n+1}$. If $K$ is given, its cardinality is $q$. If $V$ is given, it must be 4-dimensional, and over $K$. The function returns:

**(a)**
The group $G$;

**(b)**
An indexed set of the generators of the 1-dimensional subspaces of $K^{(n)}$, giving the correspondence between these vectors and the $G$-set of $G$.

## `AffineGroup(M): GrpMat[FldFin] -> GrpPerm, {@ ModTupFldElt  @}`

Given a matrix group of degree $d$ over a finite field $F$, construct the semidirect product $V:M$, where $V=F^d$ is the natural $M$-module. The result $G$ is a standard permutation group of degree $|V| = |F|^d$, where the second return value gives the correspondence between the elements of $V$ and the standard $G$-set.