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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.