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Several functions are provided which construct various classical
groups and other groups of Lie type. The effect of these functions
is to define the group in terms of a set of generating matrices.
As shown by Chevalley, for each simple Lie algebra L over the complex
field and for each finite field GF(q) there is an associated matrix
group L(q). In general, these groups are perfect but not simple. To
obtain the simple group, it is necessary to form the quotient by the centre.
Similarly, as Steinberg, Ree and others have shown, if the associated
Coxeter graph has an automorphism, of order t say, then there will
be a `twisted' version sL(q) of L(q).
Generators for the series A, C, ()2A and ()2B are described in
[Tay87]. Generators for the series B, D and ()2D are as given
by Rylands and Taylor [RT98]. Generators for the exceptional
groups of Lie type are described by Howlett, Rylands and Taylor in
[HRT01].
ChevalleyGroup(X, n, q: parameters) : MonStgElt, RngIntElt, RngIntElt -> GrpMat
Irreducible: BoolElt Default: false
Construct a matrix group over the field K (or over GF(q)) which has the
adjoint Chevalley group of Lie series X and Lie rank n as the quotient
modulo scalar matrices. In most cases the group returned is the universal Chevalley
group Xn(q); however, for series B, D and 2D the universal group
is the spin group and the matrix group returned by ChevalleyGroup is
Ω(2n + 1, q), Ω^ + (2n, q) or Ω^ - (2n, q).
For the twisted groups the meaning of the parameter q is consistent with the
(abbreviated) notation in the `Atlas of Finite Groups' and in the
monograph series `The Classification of the Finite Simple Groups' by Gorenstein,
Lyons and Solomon. For a Chevalley group of rank n and type X with an
automorphism of order t the Atlas defines the twisted Chevalley group
()tXn(q, qt) to be the set of elements of Xn(qt) fixed by the quotient
of the twisting automorphism and the field automorphism induced by
x |-> xq of GF(qt). In the Atlas the abbreviated
notation for the twisted group is ()tXn(q) but in Carter [Car72] it
is ()tXn(qt). The first signature of the intrinsic expects the field
GF(qt) but the second signature expects the parameter q.
For example, for the series "2A", the group 2A_n(q)
is SU(n + 1, q) but, in the first form of the signature, K must be the field
GF(q2). Similarly the first form of the signature for the groups ()3D4(q) and
()2E6(q) requires the fields GF(q3) and GF(q2), respectively.
The possible series and the groups returned are:
- "A": n≥0, An(q), the special linear group SL(n + 1, q).
- "B": n≥1, Bn(q), the orthogonal group Ω(2n + 1, q).
- "C": n≥1, Cn(q), the symplectic group Sp(2n, q).
- "D": n≥1, Dn(q), the orthogonal group D_n(q) = Omega+(2n, q).
- "E": n∈{ 6, 7, 8 }, the exceptional groups En(q).
E6(q) is represented as a matrix group of degree 27. It is simple
unless q ≡ 1 mod 3, in which case its centre has order 3.
E7(q) is represented as a matrix group of degree 56. It is simple
unless q ≡ 1 mod 2, in which case its centre has order 2.
E8(q) is represented as a matrix group of degree 248.
- "F": n = 4, the exceptional group F4(q) represented as a matrix
group of degree 26. If q = 3k then this representation is reducible.
An irreducible representation is not yet available.
- "G": n = 2, the exceptional group G2(q) represented as a matrix
group of degree 7. If q = 2k then this representation is reducible.
An irreducible representation of degree 6 can be obtained by setting the
parameter Irreducible := true.
- "2A": n≥1, K = GF(q2), the special unitary group 2A_n(q) = SU(n + 1, q).
- "2B": n = 2, K = GF(q), q = 22k + 1, the Suzuki group 2B_2(q) = Sz(q).
- "2D": n≥1, K =GF(q), ()2Dn(q), the orthogonal group 2D_n(q) = Omega-(2n, q).
- "3D": n = 4, K = GF(q3), the exceptional group 3D_4(q).
- "2E": n = 6, K = GF(q2), the exceptional group 2E_6(q).
- "2F": n = 4, K=GF(q), q = 22k + 1, the Ree group 2F_4(q), simple
except when q = 2 when the derived group is simple and is returned by the
function TitsGroup.
- "2G": n = 2, K = GF(q), q = 32k + 1, the Ree
group 2G_2(q), simple except when q = 3.
The orders of the universal Chevalley groups Xn(q) and ()tXn(q) are
polynomials in q. For the twisted groups of types ()2An, ()3D4 and
()2E6 the parameter q is the order of the fixed field of the Frobenius
automorphism.
Other versions of Chevalley groups are quotients of universal Chevalley groups
modulo a subgroup of the centre.
FactoredChevalleyGroupOrder(type, n, q: parameters) : MonStgElt, RngIntElt, RngIntElt -> RngIntEltFact
Proof: BoolElt Default: true
Version: MonStgElt Default: em "Default"
ChevalleyGroupOrder(type, n, F: parameters) : MonStgElt, RngIntElt, FldFin -> RngIntEltFact
ChevalleyGroupOrder(type, n, q: parameters) : MonStgElt, RngIntElt, RngIntElt -> RngIntEltFact
Version: MonStgElt Default: em "Default"
The (factored) order of the Chevalley group of a given type and rank
over the field F (or GF(q)). The default is the order of the group returned by
ChevalleyGroup, which except for types Bn, Dn and ()2Dn is the
universal group. The orders of the universal and adjoint Chevalley group can be obtained
by setting the parameter Version to Universal or Adjoint. In
the factored version the value of Proof is passed to the Magma's factorisation
function (q.v.).
Magma offers several functions to construct the classical groups.
For most of these functions, it is possible to specify the particular
group by giving one of the following combinations of arguments:
- (i)
- The degree n and the coefficient field K of the desired
matrix group;
- (ii)
- The degree n of the desired matrix group and a prime power q
which relates the group to the appropriate Lie algebra. With the exception of
the unitary groups (which will be defined over GF(q2)),
the resulting group will be defined over GF(q); or,
- (iii)
- A full vector space V = Kn on which the desired matrix group
should act naturally.
GeneralLinearGroup(n, q) : RngIntElt, RngIntElt -> GrpMat
GeneralLinearGroup(n, K) : RngIntElt, FldFin -> GrpMat
GeneralLinearGroup(V) : ModTupRng -> GrpMat
GL(n, q) : RngIntElt, RngIntElt -> GrpMat
GL(n, K) : RngIntElt, FldFin -> GrpMat
GL(V) : ModTupRng -> GrpMat
Here n is a positive integer, q is the power of a prime,
K is a finite field GF(q), and V is an n-dimensional
vector space over K. This function constructs the general
linear group GL(n, q) (resp. GL(n, K), GL(V)) in terms
of generating matrices. The intrinsic name may be abbreviated
to GL.
SpecialLinearGroup(n, K) : RngIntElt, FldFin -> GrpMat
SpecialLinearGroup(V) : ModTupRng -> GrpMat
SL(n, q) : RngIntElt, RngIntElt -> GrpMat
SL(n, K) : RngIntElt, FldFin -> GrpMat
SL(V) : ModTupRng -> GrpMat
Here n is a positive integer, q is the power of a prime,
K is a finite field GF(q), and V is an n-dimensional
vector space over K. This function constructs the
special linear group SL(n, q) (resp. SL(n, K), SL(V)),
namely the group of n x n matrices of determinant 1,
in terms of generating matrices. The intrinsic name may be
abbreviated to SL.
AffineGeneralLinearGroup(GrpMat, n, K) : Cat, RngIntElt, FldFin -> GrpMat
AffineGeneralLinearGroup(GrpMat, V) : Cat, ModTupRng -> GrpMat
AffineGeneralLinearGroup(E) : GrpPerm -> GrpPerm
AGL(GrpMat, n, q) : Cat, RngIntElt, RngIntElt -> GrpMat
AGL(GrpMat, n, K) : Cat, RngIntElt, FldFin -> GrpMat
AGL(GrpMat, V) : Cat, ModTupRng -> GrpMat
AGL(E) : GrpPerm -> GrpPerm
Here n is a positive integer greater than or equal to 2, q is the
power of a prime, K is a finite field GF(q), and V is an
n-dimensional vector space over K. This function constructs
the affine general linear group AGL(n, q) (resp. AGL(n, K),
AGL(V)) as a subgroup of GL(n + 1, K). If the category name
GrpMat is omitted the affine group will be returned as a
permutation group. The intrinsic name may be abbreviated to AGL.
If E is a regular elementary abelian permutation group the function
constructs the affine general linear group of E, regarded
as a vector space over the field GF(p), where p is the exponent
of E.
AffineSpecialLinearGroup(GrpMat, n, K) : Cat, RngIntElt, FldFin -> GrpMat
AffineSpecialLinearGroup(GrpMat, V) : Cat, ModTupRng -> GrpMat
ASL(GrpMat, n, q) : Cat, RngIntElt, RngIntElt -> GrpMat
ASL(GrpMat, n, K) : Cat, RngIntElt, FldFin -> GrpMat
ASL(GrpMat, V) : Cat, ModTupRng -> GrpMat
Here n is a positive integer greater than or equal to 2, q is the
power of a prime, K is a finite field GF(q), and V is an
n-dimensional vector space over K. This function constructs
the affine special linear group ASL(n, q) (resp. ASL(n, K),
ASL(V)) as a subgroup of SL(n + 1, K). If the category name
GrpMat is omitted, the affine group will be returned as a
permutation group. The intrinsic name may be abbreviated to ASL.
ConformalUnitaryGroup(n, K) : RngIntElt, FldFin -> GrpMat
ConformalUnitaryGroup(V): ModTupRng -> GrpMat
CU(n, q) : RngIntElt, RngIntElt -> GrpMat
CU(n, K) : RngIntElt, FldFin -> GrpMat
CU(V): ModTupRng -> GrpMat
Here n ≥2 is a positive integer, q is the power of a prime,
K is the finite field GF(q2), and V is the n-dimensional
vector space over K. This function constructs the conformal unitary
group CU(n, q) (resp. CU(n, K), CU(V)) in terms of
generating matrices. It preserves the standard hermitian form up to
a non-zero scalar multiple. The intrinsic name may be abbreviated
to CU.
GeneralUnitaryGroup(n, K) : RngIntElt, FldFin -> GrpMat
GeneralUnitaryGroup(V): ModTupRng -> GrpMat
GU(n, q) : RngIntElt, RngIntElt -> GrpMat
GU(n, K) : RngIntElt, FldFin -> GrpMat
GU(V): ModTupRng -> GrpMat
Here n ≥2 is a positive integer, q is the power of a prime,
K is the finite field GF(q2), and V is the n-dimensional
vector space over K. This function constructs the general unitary
group GU(n, q) (resp. GU(n, K), GU(V)) in terms of
generating matrices. It preserves the standard hermitian form.
The intrinsic name may be abbreviated to GU.
SpecialUnitaryGroup(n, K) : RngIntElt, FldFin -> GrpMat
SpecialUnitaryGroup(V): ModTupRng -> GrpMat
SU(n, q) : RngIntElt, RngIntElt -> GrpMat
SU(n, K) : RngIntElt, FldFin -> GrpMat
SU(V): ModTupRng -> GrpMat
Here n is an integer greater than or equal to 2, q is the power
of a prime, K is the finite field GF(q2), and V is the
n-dimensional vector space over K. This function constructs
the special unitary group SU(n, q) (resp. SU(n, K), SU(V))
in terms of generating matrices. The intrinsic name may be abbreviated
to SU.
CSU(n, q) : RngIntElt, RngIntElt -> GrpMat
For an integer n≥2 and a prime power q, this function
constructs the normaliser of the special unitary group SU(n, q)
in SL(n, q2). The intrinsic name may be abbreviated to CSU.
ConformalSymplecticGroup(n, K) : RngIntElt, FldFin -> GrpMat
ConformalSymplecticGroup(V) : ModTupRng -> GrpMat
CSp(n, q) : RngIntElt, RngIntElt -> GrpMat
CSp(n, K) : RngIntElt, FldFin -> GrpMat
CSp(V) : ModTupRng -> GrpMat
Here n is an even integer greater than or equal to 4, q is the
power of a prime, K is the finite field GF(q), and V is the
n-dimensional vector space over K. This function constructs
the conformal symplectic group CSp(n, q) (resp. CSp(n, K), CSp(V))
in terms of generating matrices. It preserves the standard alternating
form up to a non-zero scalar multiple. The intrinsic name may be abbreviated
to CSp.
SymplecticGroup(n, K) : RngIntElt, FldFin -> GrpMat
SymplecticGroup(V) : ModTupRng -> GrpMat
Sp(n, q) : RngIntElt, RngIntElt -> GrpMat
Sp(n, K) : RngIntElt, FldFin -> GrpMat
Sp(V) : ModTupRng -> GrpMat
Here n is an even integer greater than or equal to 4, q is the
power of a prime, K is the finite field GF(q), and V is the
n-dimensional vector space over K. This function constructs
the symplectic group Sp(n, q) (resp. Sp(n, K), Sp(V))
in terms of generating matrices. It preserves the standard alternating
form. The intrinsic name may be abbreviated to Sp.
ConformalOrthogonalGroup(n, K) : RngIntElt, FldFin -> GrpMat
ConformalOrthogonalGroup(V): ModTupRng -> GrpMat
CO(n, q) : RngIntElt, RngIntElt -> GrpMat
CO(n, K) : RngIntElt, FldFin -> GrpMat
CO(V): ModTupRng -> GrpMat
Here n is an odd integer greater than or equal to 3, q is the
power of a prime, K is the finite field GF(q), and V is the
n-dimensional vector space over K. This function constructs
the conformal orthogonal group CO(n, q) (resp. CO(n, K), CO(V))
in terms of generating matrices. It preserves the standard quadratic
form up to a non-zero scalar multiple. The intrinsic name may be
abbreviated to CO.
GeneralOrthogonalGroup(n, K) : RngIntElt, FldFin -> GrpMat
GeneralOrthogonalGroup(V): ModTupRng -> GrpMat
GO(n, q) : RngIntElt, RngIntElt -> GrpMat
GO(n, K) : RngIntElt, FldFin -> GrpMat
GO(V): ModTupRng -> GrpMat
Here n is an odd integer greater than or equal to 3, q is the
power of a prime, K is the finite field GF(q), and V is the
n-dimensional vector space over K. This function constructs
the general orthogonal group GO(n, q) (resp. GO(n, K), GO(V))
in terms of generating matrices. It preserves the standard quadratic form.
The intrinsic name may be abbreviated to GO.
SpecialOrthogonalGroup(n, K) : RngIntElt, FldFin -> GrpMat
SpecialOrthogonalGroup(V): ModTupRng -> GrpMat
SO(n, q) : RngIntElt, RngIntElt -> GrpMat
SO(n, K) : RngIntElt, FldFin -> GrpMat
SO(V): ModTupRng -> GrpMat
Here n is an odd integer greater than or equal to 3, q is the
power of a prime, K is the finite field GF(q), and V is the
n-dimensional vector space over K. This function constructs
the special orthogonal group SO(n, q) (resp. SO(n, K),
SO(V)) in terms of generating matrices. The intrinsic name
may be abbreviated to SO. In characteristic 2 the group coincides
with the general orthogonal group.
CSO(n, q) : RngIntElt, RngIntElt -> GrpMat
For an odd integer n≥3 and a prime power q, this function
constructs the normaliser of the special orthogonal group SO(n, q)
in SL(n, q). The intrinsic name may be abbreviated to CSO.
ConformalOrthogonalGroupPlus(n, K) : RngIntElt, FldFin -> GrpMat
ConformalOrthogonalGroupPlus(V): ModTupRng -> GrpMat
COPlus(n, q) : RngIntElt, RngIntElt -> GrpMat
COPlus(n, K) : RngIntElt, FldFin -> GrpMat
COPlus(V): ModTupRng -> GrpMat
Here n is an even integer greater than or equal to 2, q is the
power of a prime, K is the finite field GF(q), and V is the
n-dimensional vector space over K. This function constructs
the conformal orthogonal group CO^ + (n, q) (resp. CO^ + (n, K), CO^ + (V))
in terms of generating matrices. It preserves the standard quadratic form
up to a non-zero scalar multiple. The intrinsic name may be
abbreviated to COPlus.
GeneralOrthogonalGroupPlus(n, K) : RngIntElt, FldFin -> GrpMat
GeneralOrthogonalGroupPlus(V): ModTupRng -> GrpMat
GOPlus(n, q) : RngIntElt, RngIntElt -> GrpMat
GOPlus(n, K) : RngIntElt, FldFin -> GrpMat
GOPlus(V): ModTupRng -> GrpMat
Here n is an even integer greater than or equal to 2, q is the
power of a prime, K is the finite field GF(q), and V is the
n-dimensional vector space over K. This function constructs
the general orthogonal group GO^ + (n, q) (resp. GO^ + (n, K),
GO^ + (V)) in terms of generating matrices. It preserves the standard
quadratic form. The intrinsic name may be abbreviated to GOPlus.
SpecialOrthogonalGroupPlus(n, K) : RngIntElt, FldFin -> GrpMat
SpecialOrthogonalGroupPlus(V): ModTupRng -> GrpMat
SOPlus(n, q) : RngIntElt, RngIntElt -> GrpMat
SOPlus(n, K) : RngIntElt, FldFin -> GrpMat
SOPlus(V): ModTupRng -> GrpMat
Here n is an even integer greater than or equal to 2, q is the
power of a prime, K is the finite field GF(q), and V is the
n-dimensional vector space over K. This function constructs
the special orthogonal group SO^ + (n, q) (resp. SO^ + (n, K),
SO^ + (V)) in terms of generating matrices. The intrinsic name
may be abbreviated to SOPlus. In characteristic 2 the group coincides
with GOPlus and is not the kernel of the Dickson invariant.
CSOPlus(n, q) : RngIntElt, RngIntElt -> GrpMat
For an odd integer n≥3 and a prime power q, this function
constructs the normaliser of the special orthogonal group SO^ + (n, q)
in SL(n, q). The intrinsic name may be abbreviated to CSOPlus.
ConformalOrthogonalGroupMinus(n, K) : RngIntElt, FldFin -> GrpMat
ConformalOrthogonalGroupMinus(V): ModTupRng -> GrpMat
COMinus(n, q) : RngIntElt, RngIntElt -> GrpMat
COMinus(n, K) : RngIntElt, FldFin -> GrpMat
COMinus(V): ModTupRng -> GrpMat
Here n is an even integer greater than or equal to 2, q is the
power of a prime, K is the finite field GF(q), and V is the
n-dimensional vector space over K. This function constructs
the conformal orthogonal group CO^ - (n, q) (resp. CO^ - (n, K), CO^ - (V))
in terms of generating matrices. It preserves the standard quadratic form
(of Minus type) up to a non-zero scalar multiple. The intrinsic name may be
abbreviated to COMinus.
GeneralOrthogonalGroupMinus(n, K) : RngIntElt, FldFin -> GrpMat
GeneralOrthogonalGroupMinus(V): ModTupRng -> GrpMat
GOMinus(n, q) : RngIntElt, RngIntElt -> GrpMat
GOMinus(n, K) : RngIntElt, FldFin -> GrpMat
GOMinus(V): ModTupRng -> GrpMat
Here n is an even integer greater than or equal to 2, q is the
power of a prime, K is the finite field GF(q), and V is the
n-dimensional vector space over K. This function constructs
the general orthogonal group GO^ - (n, q) (resp. GO^ - (n, K),
GO^ - (V)) in terms of generating matrices. It preserves the standard
quadratic form (of Minus type). The intrinsic name may be abbreviated
to GOMinus.
SpecialOrthogonalGroupMinus(n, K) : RngIntElt, FldFin -> GrpMat
SpecialOrthogonalGroupMinus(V): ModTupRng -> GrpMat
SOMinus(n, q) : RngIntElt, RngIntElt -> GrpMat
SOMinus(n, K) : RngIntElt, FldFin -> GrpMat
SOMinus(V): ModTupRng -> GrpMat
Here n is an even integer greater than or equal to 2, q is the
power of a prime, K is the finite field GF(q), and V is the
n-dimensional vector space over K. This function constructs
the special orthogonal group SO^ - (n, q) (resp. SO^ - (n, K),
SO^ - (V)) in terms of generating matrices. The intrinsic name
may be abbreviated to SOMinus. In characteristic 2 the group coincides
with GOMinus and is not the kernel of the Dickson invariant.
CSOMinus(n, q) : RngIntElt, RngIntElt -> GrpMat
For an odd integer n≥3 and a prime power q, this function
constructs the normaliser of the special orthogonal group SO^ - (n, q)
in SL(n, q). The intrinsic name may be abbreviated to CSOMinus.
Omega(n, K) : RngIntElt, FldFin -> GrpMat
Omega(V): ModTupRng -> GrpMat
Here n is an odd integer greater than or equal to 3, q is a
power of a prime, K is the finite field GF(q), and V is the
n-dimensional vector space over K. This function constructs
the orthogonal group Ω(n, K) (resp. Ω(n, K),
Ω(V)) in terms of two generating matrices. If q is odd, the
group Ω(n, K) is the kernel of the spinor norm map on SO(n, K));
if q is even, it is the kernel of the Dickson invariant.
OmegaPlus(n, K) : RngIntElt, FldFin -> GrpMat
OmegaPlus(V): ModTupRng -> GrpMat
Here n is an even integer greater than or equal to 2, q is a
power of a prime, K is the finite field GF(q), and V is the
n-dimensional vector space over K. This function constructs
the orthogonal group Ω^ + (n, q) (resp. Ω^ + (n, K),
Ω^ + (V)) in terms of two generating matrices. If q is odd, the
group Ω^ + (n, K) is the kernel of the spinor norm map on SO^ + (n, K).
If q is even, it is the kernel of the Dickson invariant.
OmegaMinus(n, K) : RngIntElt, FldFin -> GrpMat
OmegaMinus(V): ModTupRng -> GrpMat
Here n is an even integer greater than or equal to 2, q is a
power of a prime, K is the finite field GF(q), and V is the
n-dimensional vector space over K. This function constructs
the orthogonal group Ω^ - (n, q) (resp. Ω^ - (n, K),
Ω^ - (V)) in terms of two generating matrices. If q is odd, the
group Ω^ - (n, K) is the kernel of the spinor norm map on SO^ - (n, K).
If q is even, it is the kernel of the Dickson invariant.
Spin(n, K) : RngIntElt, FldFin -> GrpMat
Spin(V): ModTupRng -> GrpMat
Here n is an odd integer greater than or equal to 1, q is the
power of a prime, K is the finite field GF(q), and V is the
n-dimensional vector space over K. This function constructs
the spin group Spin(n, K) (resp. Spin(n, K),
Spin(V)).
SpinPlus(n, K) : RngIntElt, FldFin -> GrpMat
SpinPlus(V): ModTupRng -> GrpMat
Here n is an even integer greater than or equal to 2, q is the
power of a prime, K is the finite field GF(q), and V is the
n-dimensional vector space over K. This function constructs
the spin group Spin^ + (n, K) (resp. Spin^ + (n, K),
Spin^ + (V)).
SpinMinus(n, K) : RngIntElt, FldFin -> GrpMat
SpinMinus(V): ModTupRng -> GrpMat
Here n is an even integer greater than or equal to 4, q is the
power of a prime, K is the finite field GF(q), and V is the
n-dimensional vector space over K. This function constructs
the spin group Spin^ - (n, K) (resp. Spin^ - (n, K),
Spin^ - (V)).
The Suzuki groups are specified slightly differently, as the degree of
the group is always four. Thus for this family of groups, the possible
combinations of arguments are:
- (i)
- A finite field K = GF(22m + 1), over which the resulting
matrix group is defined;
- (ii)
- An integer q = 22m + 1, corresponding to the field K = GF(q)
over which the resulting matrix group is defined; or,
- (iii)
- A vector space V = K4 where K = GF(22m + 1) on which the
resulting matrix group acts naturally.
which the resulting
SuzukiGroup(K) : FldFin -> GrpMat
SuzukiGroup(V) : ModTupRng -> GrpMat
Here q is a prime power of the form 22n + 1, K is the finite field
GF(q), and V is the 4-dimensional vector space over K. This
function constructs the Suzuki simple group Sz(q) (resp. Sz(K),
Sz(V)) in terms of two generating matrices. The intrinsic name
may be abbreviated to Sz.
We create the 10-dimensional symplectic group over GF(8):
> F<u> := FiniteField(8);
> G := SymplecticGroup(10, F);
> G;
MatrixGroup(10, GF(2, 3))
Generators:
[ u 0 0 0 0 0 0 0 0 0]
[ 0 1 0 0 0 0 0 0 0 0]
[ 0 0 1 0 0 0 0 0 0 0]
[ 0 0 0 1 0 0 0 0 0 0]
[ 0 0 0 0 u 0 0 0 0 0]
[ 0 0 0 0 0 u 0 0 0 0]
[ 0 0 0 0 0 0 1 0 0 0]
[ 0 0 0 0 0 0 0 1 0 0]
[ 0 0 0 0 0 0 0 0 1 0]
[ 0 0 0 0 0 0 0 0 0 u^6]
[0 0 0 1 1 1 0 0 0 0]
[1 0 0 0 0 0 0 0 0 0]
[0 1 0 0 0 0 0 0 0 0]
[0 0 1 0 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0 0 0]
[0 0 0 0 1 0 1 0 0 0]
[0 0 0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 0 0 1 0]
[0 0 0 0 0 0 0 0 0 1]
[0 0 0 0 1 0 0 0 0 0]
We create the Suzuki group over GF(128):
> F<w> := FiniteField(128);
> V := VectorSpace(F, 4);
> S := SuzukiGroup(V);
> S;
MatrixGroup(4, GF(2, 7))
Generators:
[0 0 0 1]
[0 0 1 0]
[0 1 0 0]
[1 0 0 0]
[ w^8 0 0 0]
[ 0 w^120 0 0]
[ 0 0 w^7 0]
[ 0 0 0 w^119]
[ 1 0 0 0]
[ w^8 1 0 0]
[ 0 w 1 0]
[w^17 w^9 w^8 1]
> Order(S);
34093383680
> FactoredOrder(S);
[ <2, 14>, <5, 1>, <29, 1>, <113, 1>, <127, 1> ]
The Ree groups ((2)G2(q)) are given in an irreducible matrix representation
of degree seven. The possible combinations of arguments are:
- (i)
- A finite field K = GF(32m + 1) with m > 0, over which the
matrix group is defined.
- (ii)
- An integer q = 32m + 1 with m > 0, corresponding to the
field K = GF(q) over which the group is defined; or,
- (iii)
- A vector space V = K7 where K = GF(32m + 1) with m > 0,
on which the matrix group acts naturally.
ReeGroup(K) : FldFin -> GrpMat
ReeGroup(V) : ModTupRng -> GrpMat
Here q is a prime power of the form q = 32m + 1 with m >
0, K is the finite field GF(q), and V is the 7-dimensional
vector space over K. This function constructs the Ree group
(2)G2(q) (resp. (2)G2(K), (2)G2(V)) in terms of standard
generating matrices. The intrinsic name may be abbreviated to Ree.
The Ree groups ((2)F4(q)) are given in an irreducible matrix representation
of degree twenty-six. The possible combinations of arguments are:
- (i)
- A finite field K = GF(22m + 1) with m > 0, over which the
matrix group is defined.
- (ii)
- An integer q = 22m + 1 with m > 0, corresponding to the
field K = GF(q) over which the group is defined; or,
- (iii)
- A vector space V = K26 where K = GF(22m + 1) with m > 0,
on which the matrix group acts naturally.
LargeReeGroup(K) : FldFin -> GrpMat
LargeReeGroup(V) : ModTupRng -> GrpMat
Here q is a prime power of the form q = 22m + 1 with m >
0, K is the finite field GF(q), and V is the 26-dimensional
vector space over K. This function constructs the Ree group
(2)F4(q) (resp. (2)F4(K), (2)F4(V)) in terms of standard
generating matrices. The intrinsic name may be abbreviated to LargeRee.
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