Each of the functions in this family returns two values:

• A permutation group G corresponding to the action of a designated matrix group M on a vector space V; and
• 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.

• Integers n and q corresponding to the degree and the field GF(q) of M (GF(q^2) in the case of the unitary groups).
• An integer n and a finite field K corresponding to the degree and the coefficient field of M.
• A vector space V = K^n on which M natrually acts.

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 = AGL(n, q), i.e., the group corresponding to the action of GL(n, q) on the affine points of the n-dimensional vector space V over K = GF(q). The function returns:

• The group G;
• An indexed set giving the correspondence between the affine points and the G-set of G.

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 = ASL(n, q), i.e., the group corresponding to the action of SL(n, q) on the affine points of the n-dimensional vector space V over K = GF(q). The function returns:

• The group G;
• An indexed set giving the correspondence between the affine points and the G-set of G.

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 = AGammaL(n, q), i.e., the group corresponding to the action of GammaL(n, q) (the automorphism group of GL(n, q)) on the affine points of the n-dimensional vector space V over K = GF(q). The function returns:

• The group G;
• An indexed set giving the correspondence between the points and the G-set of G.

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 = ASigmaL(n, q), i.e., the group corresponding to the action of SigmaL(n, q) (the automorphism group of SL(n, q)) on the affine points of the n-dimensional vector space V over K = GF(q). The function returns:

• The group G;
• An indexed set giving the correspondence between the points and the G-set of G.

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 = PGL(n, q), i.e., the group corresponding to the action of GL(n, q) on the projective points of the n-dimensional vector space V over K = GF(q), where n >= 2 and q is a prime power. The function returns:

• The group G;
• 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.

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 = PSL(n, q), i.e., the group corresponding to the action of SL(n, q) on the projective points of the n-dimensional vector space V over K = GF(q), where n >= 2 and q is a prime power. The function returns:

• The group G;
• 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.

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 the automorphism group G = PGammaL(n, q) of the projective general linear group PGL(n, q) corresponding to the n-dimensional vector space V over the field K = GF(q), where n >= 2 and q is a prime power. The function returns:

• The group G;
• An indexed set giving the correspondence between the points and the G-set of G.

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 the automorphism group G = PSigmaL(n, q) of the projective special linear group PSL(n, q) corresponding to the n-dimensional vector space V over the field K = GF(q), where n >= 2 and q is a prime power. The function returns:

• The group G;
• An indexed set giving the correspondence between the points and the G-set of G.

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 = PGU(n, q) corresponding to the n-dimensional vector space V over the field K = GF(q^2), where n >= 2 and q is a prime power. The function returns:

• The group G;
• 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.

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 = PSU(n, q) corresponding to the n-dimensional vector space V over the field K = GF(q^2), where n >= 2 and q is a prime power. The function returns:

• The group G;
• 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.

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 the automorphism group G = PGammaU(n, q) of the projective general unitary group PGU(n, q) corresponding to the n-dimensional vector space V over the field K = GF(q^2), where n >= 2 and q is a prime power. The function returns:

• The group G;
• An indexed set giving the correspondence between the points and the G-set of G.

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 = PSigmaU(n, q) of the projective special unitary group PSU(n, q) corresponding to the n-dimensional vector space V over the field K = GF(q^2), where n >= 2 and q is a prime power. The function returns:

• The group G;
• An indexed set giving the correspondence between the points and the G-set of G.

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 = PSp(n, q), where K = GF(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:

• The group G;
• 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.

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 automorphism group G = PSigmaSp(n, q) of the projective symplectic group PSp(n, q), where K = GF(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:

• The group G;
• An indexed set giving the correspondence between the points and the G-set of G.

ProjectiveGeneralOrthogonalGroup(n, q) : RngIntElt, RngIntElt -> GrpMat

ProjectiveGeneralOrthogonalGroup(n, K) : RngIntElt, FldFin -> GrpMat

ProjectiveGeneralOrthogonalGroup(V): ModTupRng -> GrpMat

PGO(n, q) : RngIntElt, RngIntElt -> GrpMat

PGO(n, K) : RngIntElt, FldFin -> GrpMat

PGO(V): ModTupRng -> GrpMat

Construct the projective general orthogonal group G = PGO(n, q), where K = GF(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:

• The group G;
• 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(n, q) : RngIntElt, RngIntElt -> GrpMat

ProjectiveGeneralOrthogonalGroupPlus(n, K) : RngIntElt, FldFin -> GrpMat

ProjectiveGeneralOrthogonalGroupPlus(V): ModTupRng -> GrpMat

PGOPlus(n, q) : RngIntElt, RngIntElt -> GrpMat

PGOPlus(n, K) : RngIntElt, FldFin -> GrpMat

PGOPlus(V): ModTupRng -> GrpMat

Construct the projective general orthogonal group G = PGO^ + (n, q), where K = GF(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:

• The group G;
• 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(n, q) : RngIntElt, RngIntElt -> GrpMat

ProjectiveGeneralOrthogonalGroupMinus(n, K) : RngIntElt, FldFin -> GrpMat

ProjectiveGeneralOrthogonalGroupMinus(V): ModTupRng -> GrpMat

PGOMinus(n, q) : RngIntElt, RngIntElt -> GrpMat

PGOMinus(n, K) : RngIntElt, FldFin -> GrpMat

PGOMinus(V): ModTupRng -> GrpMat

Construct the projective general orthogonal group G = PGO^ - (n, q), where K = GF(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:

• The group G;
• 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(n, q) : RngIntElt, RngIntElt -> GrpMat

ProjectiveSpecialOrthogonalGroup(n, K) : RngIntElt, FldFin -> GrpMat

ProjectiveSpecialOrthogonalGroup(V): ModTupRng -> GrpMat

PSO(n, q) : RngIntElt, RngIntElt -> GrpMat

PSO(n, K) : RngIntElt, FldFin -> GrpMat

PSO(V): ModTupRng -> GrpMat

Construct the projective special orthogonal group G = PSO(n, q), where K = GF(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:

• The group G;
• 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(n, q) : RngIntElt, RngIntElt -> GrpMat

ProjectiveSpecialOrthogonalGroupPlus(n, K) : RngIntElt, FldFin -> GrpMat

ProjectiveSpecialOrthogonalGroupPlus(V): ModTupRng -> GrpMat

PSOPlus(n, q) : RngIntElt, RngIntElt -> GrpMat

PSOPlus(n, K) : RngIntElt, FldFin -> GrpMat

PSOPlus(V): ModTupRng -> GrpMat

Construct the projective special orthogonal group G = PSO^ + (n, q), where K = GF(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:

• The group G;
• 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(n, q) : RngIntElt, RngIntElt -> GrpMat

ProjectiveSpecialOrthogonalGroupMinus(n, K) : RngIntElt, FldFin -> GrpMat

ProjectiveSpecialOrthogonalGroupMinus(V): ModTupRng -> GrpMat

PSOMinus(n, q) : RngIntElt, RngIntElt -> GrpMat

PSOMinus(n, K) : RngIntElt, FldFin -> GrpMat

PSOMinus(V): ModTupRng -> GrpMat

Construct the projective general orthogonal group G = PSO^ - (n, q), where K = GF(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:

• The group G;
• 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.

POmega(arguments)

ProjectiveOmega(n, q) : RngIntElt, RngIntElt -> GrpMat

ProjectiveOmega(n, K) : RngIntElt, FldFin -> GrpMat

ProjectiveOmega(V): ModTupRng -> GrpMat

POmega(n, q) : RngIntElt, RngIntElt -> GrpMat

POmega(n, K) : RngIntElt, FldFin -> GrpMat

POmega(V): ModTupRng -> GrpMat

Construct the projective orthogonal group G = POmega(n, q), where K = GF(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:

• The group G;
• 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.

POmegaPlus(arguments)

ProjectiveOmegaPlus(n, q) : RngIntElt, RngIntElt -> GrpMat

ProjectiveOmegaPlus(n, K) : RngIntElt, FldFin -> GrpMat

ProjectiveOmegaPlus(V): ModTupRng -> GrpMat

POmegaPlus(n, q) : RngIntElt, RngIntElt -> GrpMat

POmegaPlus(n, K) : RngIntElt, FldFin -> GrpMat

POmegaPlus(V): ModTupRng -> GrpMat

Construct the projective orthogonal group G = POmega(n, q), where K = GF(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:

• The group G;
• 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.

POmegaMinus(arguments)

ProjectiveOmegaMinus(n, q) : RngIntElt, RngIntElt -> GrpMat

ProjectiveOmegaMinus(n, K) : RngIntElt, FldFin -> GrpMat

ProjectiveOmegaMinus(V): ModTupRng -> GrpMat

POmegaMinus(n, q) : RngIntElt, RngIntElt -> GrpMat

POmegaMinus(n, K) : RngIntElt, FldFin -> GrpMat

POmegaMinus(V): ModTupRng -> GrpMat

Construct the projective orthogonal group G = POmega(n, q), where K = GF(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:

• The group G;
• 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.

PSz(arguments)

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

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

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

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

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

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

Construct the permutation representation G = PSz(q) of the Suzuki simple group 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:

• The group G;
• 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.