Permutation representation of a finitely presented group: 

If is grelgroup with relations then :=subgrel({}, ) is the identity subgroup of . Since permrep() is a permutation representation of a homomorphic image of by , thus it is a permutation group isomorphic to . 

Examples: 

>	g := grelgroup({a, b}, {[a, a, a], [b, b], [a, b, 1/a, 1/b]}); sg := subgrel({y = ([b, b])}, g); group:-permrep(sg);

 

(1.1.1)

 

General Linear Groups: 

The general linear group GL(n,q) is a set of all n×n invertible matrices with entries in  

• GeneralLinearGroup(n,q)

• GeneralLinearGroupOrder(n,q)

This function returns a permutation representation of the general linear group GL(n,q) where 100 for n = 2, 

20 for = 3, 10 for = 4, for = 5 and = 2 for = 6, 7, 8, 9, 10.  

Example: 

>	GeneralLinearGroup(3, 3);

 

(3.1.1)

 

>	GeneralLinearGroupOrder(3, 3);

 

(3.1.2)

 

Projective General Linear Groups: 

If G = GL(n,q) then the projective general linear group PGL(n,q) is a group isomorphic to the factor group G/Z(G). 

• ProjectiveGeneralLinearGroup(n,q)

• ProjectiveGeneralLinearGroupOrder(n,q)

This function returns a permutation representation of the projective general linear group PGL(n,q) where 100 for n = 2, 

20 for = 3, 10 for = 4, for = 5 and = 2 for = 6, 7, 8, 9, 10.  

Example: 

>	ProjectiveGeneralLinearGroup(4, 3);

 

(3.2.1)

 

>	ProjectiveGeneralLinearGroupOrder(4, 3);

 

(3.2.2)

 

General Unitary Groups: 

A square matrix U is called unitary if UU where U is the conjugate transpose of the matrix U. The general unitary group GU(n,q) is the set of all n×n unitary matrices with entries in .  

• GeneralUnitaryGroup(n,q)

• GeneralUnitaryGroupOrder(n,q)

This function returns a permutation representation of the general linear group GU(n,q) where 20 for n = 2, 

5 for = 3, 4 for = 4 and = 2 for = 5, 6.  

Example: 

>	GeneralUnitaryGroup(3, 2);

 

(3.3.1)

 

>	GeneralUnitaryGroupOrder(3, 2);

 

(3.3.2)

 

Projective General Unitary Groups: 

If G = GU(n,q) then the projective general linear group PGU(n,q) is a group isomorphic to the factor group G/Z(G). 

• ProjectiveGeneralUnitaryGroup(n,q)

• ProjectiveGeneralUnitaryGroupOrder(n,q)

This function returns a permutation representation of the general unitary group PGU(n,q) where 100 for n = 2, 

5 for = 3, 4 for = 4 and = 2 for = 5, 6.  

Example: 

>	ProjectiveGeneralUnitaryGroup(5, 2);

 

(3.4.1)

 

>	ProjectiveGeneralUnitaryGroupOrder(5, 2);

 

(3.4.2)

 

Special Linear Groups: 

The special linear group SL(n,q) is a set of all n×n invertible matrices with entries in and determinant 1. 

• SpecialLinearGroup(n,q)

• SpecialLinearGroupOrder(n,q)

This function returns a permutation representation of the special linear group SL(n,q) where 100 for n = 2, 

20 for = 3, 10 for = 4, for = 5 and = 2 for = 6, 7, 8, 9, 10.  

Example: 

>	SpecialLinearGroup(2, 11);

 

(3.5.1)

 

>	SpecialLinearGroupOrder(2, 11);

 

(3.5.2)

 

Special Unitary Groups: 

The special unitary group SU(n,q) is the set of all n×n unitary matrices with entries in F and determinant 1. 

• SpecialUnitaryGroup(n,q)

• SpecialUnitaryGroupOrder(n,q)

This function returns a permutation representation of the special unitary group SU(n,q) where 100 for n = 2, 

10 for = 3, 4 for = 4, for = 5 and = 2 for = 6, 7.  

Example: 

>	SpecialUnitaryGroup(4, 4);

 

(3.6.1)

 

>	SpecialUnitaryGroupOrder(4, 4);

 

(3.6.2)

 

Symplectic Groups: 

For an even number n the symplectic group Sp(n,q) is the set of all n×n matrices with entries in which preserve a nondegenerate antisymmetric bilinear form.  

• SymplecticGroup(n,q)

• SymplecticGroupOrder(n,q)

This function returns a permutation representation of the symplectic group Sp(n,q) where 100 for n = 2, 

10 for = 4, for = 6 and = 2 for = 8, 10.  

Example: 

>	SymplecticGroup(4, 3);

 

(3.7.1)

 

>	SymplecticGroupOrder(4, 3);

 

(3.7.2)

 

General Orthogonal Groups: 

A square matrix Q is called orthogonal if QQ . The general orthogonal group GO(d,n,q) is the group of n×n orthogonal matrices with entries from F. This is a subgroup of the general linear group GL(n,q). This is the group of the matrices which preserves a non-singular quadratic form over F. If the dimension n is even then d = ±1 and if n is odd then d = 0. 

• GeneralOrthogonalGroup(d,n,q)

• GeneralOrthogonalGroupOrder(d,n,q)

This function returns a permutation representation of the general orthogonal group GO(d,n,q) where 100 for n = 2, 

20 for = 3, 10 for = 4, for = 5, for n = 6, 7, 8 and = 2 for = 9, 10, 11.  

Example: 

>	GeneralOrthogonalGroup(0, 3, 11);

 

(3.8.1)

 

>	GeneralOrthogonalGroupOrder(0, 3, 11);

 

(3.8.2)

 

Special Orthogonal Groups: 

The special orthogonal group SO(d,n,q) is a subgroup of GO(d,n,q) of matrices with determinant 1. 

• SpecialOrthogonalGroup(d,n,q)

• SpecialOrthogonalGroupOrder(d,n,q)

This function returns a permutation representation of the special orthogonal group SO(d,n,q) where 100 for n = 2, 

20 for = 3, 10 for = 4, for = 5, for n = 6, 7, 8 and = 2 for = 9, 10, 11.  

Example: 

>	SpecialOrthogonalGroup(-1, 6, 3);

 

(3.9.1)

 

>	SpecialOrthogonalGroupOrder(-1, 6, 3);

 

(3.9.2)

 

Alternating Groups: 

An alternating group is the group of even permutations of a finite set. The alternating group on the set {1, ... ,n} is called the alternating group of degree n. 

• AlternatingGroup(n)

This function returns the alternating group of degree n. 

Example: 

>	AlternatingGroup(7);

 

(4.1.1)

 

Classical Groups: 

In this section we introduce the following functions: 

                   ProjectiveSpecialLinearGroup    

                   ProjectiveSpecialLinearGroupOrder                                

                   ProjectiveSpecialUnitaryGroup    

                   ProjectiveSpecialUnitaryGroupOrder                              

                   ProjectiveSymplecticGroup        

                   ProjectiveSymplecticGroupOrder                                

                   OrthogonalGroup  

                   OrthogonalGroupOrder 

Exceptional Groups: 

These are finite groups of Lie type. These are divided to two subclasses: 

1) Untwisted groups which contains and  

2) Twisted groups which contains Ree groups Suzuki groups  

and the Tits group  

• ExceptionalGroup(`name`)

• ExceptionalGroupOrder(`name`)

This function returns the exceptional group name of Lie type where name is one of the groups G2(2), G2(3), G2(4), G2(5), R(3), R(27), Sz(8), Sz(32) and 3D4(2).   

Example: 

>	ExceptionalGroup(`Sz(8)`);

 

(4.3.1)

 

>	ExceptionalGroupOrder(`Sz(8)`);

 

Sporadic Groups: 

The sporadic groups are the 26 finite simple groups that do not fit into any of the infinite families of finite simple groups. 

SporadicGroup(`name`) 

SporadicGroupOrder(`name`) 

returns the sporadic simple group name where name is in {J1, J2, J3, M11, M12, M22, M23, M24, HS, McL, HE, Co1, Co2, Co3, Suz, Fi22, Fi23, Fi24, O'N, Ru}, where these group are 

          M11, M12, M22, M23, M24                        Mathieu groups 

          J1, J2, J3                                                     Janko groups 

          HS                                                                Higman-Sims group 

          McL                                                            McLaughlin group 

          HE                                                               Held group 

          Co1, Co2, Co3                                            Conway groups 

          Suz                                                               Suzuki group 

          Fi22, Fi23, Fi24'                                          Fischer groups 

          O'N                                                              O'nan group 

          Ru                                                                Rudvalis group 

          T                                                                   Tits group 

Example: 

>	SporadicGroup(`Co3`);

 

(4.4.1)

 

>	SporadicGroupOrder(`Co3`);

 

(4.4.2)

 

Cyclic Groups: 

A cyclic group is a group that can be generated by a single element.  

• CyclicGroup(n)

• CyclicGroup(n, style=permgroup)

returns a cyclic group of order n. 

Example: 

>	CyclicGroup(19);

 

(5.1.1)

 

>	CyclicGroup(11, style=permgroup);

 

(5.1.2)

 

Symmetric Groups: 

A symmetric group is the group of all permutations of a finite set. The symmetric group on the set {1, ... , n} is called the symmetric group of degree n. 

• SymmetricGroup(n)

This function returns the symmetric group of degree n. 

Example: 

>	SymmetricGroup(14);

 

(5.2.1)

 

Dihedral Groups: 

The dihedral group Dn of order 2n is the symmetry group of an n-sided regular polygon for n > 1. 

• DihedralGroup(n)

• DihedralGroup(n, style=permgroup)

returns a dihedral group of order 2n. 

Example: 

>	DihedralGroup(7);

 

(5.3.1)

 

>	DihedralGroup(4, style=permgroup);

 

(5.3.2)

 

Dicyclic Groups: 

For each n > 1, the dicyclic group G is a non-abelian group of order 4n which contains a cyclic normal subgroup of size 2n. 

• DicyclicGroup(n)

• DicyclicGroup(n, style=permgroup)

returns a dicyclic group of order 4n. 

Example: 

>	DicyclicGroup(7);

 

(5.4.1)

 

>	DicyclicGroup(4, style=permgroup);

 

(5.4.2)

 

Metacyclic Groups: 

A group G is metacyclic if it has a cyclic normal subgroup L such that the quotient group G/L is also cyclic. The metacyclic groups are all generated by two elements which are subject to three relations depending on several numerical parameters. As a special, consider the groups generated by two elements a and b such that 

, where (mod n). 

• MetacyclicGroup(n, m, p)

• MetacyclicGroup(n, m, p, style=permgroup)

returns a metacyclic group with parameters n, m and p defined above. 

Example: 

>	MetacyclicGroup(3,2,2);

 

(5.5.1)

 

>	MetacyclicGroup(3,3,2, style=permgroup);

 

(5.5.2)