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.