Magma V2.20-8 Mon Oct 6 2014 20:49:01 on mathcompprd01 [Seed = 2549268525]
Type ? for help. Type -D to quit.
Loading startup file "/home/eobr007/.magma.startup"
> load "code.m";
Loading "code.m"
> // SetVerbose ("Classes", 1);
> // SetVerbose ("RandomSchreier", 1);
> // load "z.m";
> // SetEchoInput (true);
>
> // is vector space large enough?
> CanHaveRegularOrbit := function (G)
> d := Degree (G);
> F := BaseRing (G);
> V := VectorSpace (F, d);
> o := #G;
> if #V lt #G then "Vector space too small -- no regular orbit";
> return false, true;
> else
> return true, true;
> end if;
> end function;
>
> // determine if G has regular orbit by covering vector space
> HasRegularOrbit := function (G: Limit := 1000)
> d := Degree (G);
> F := BaseRing (G);
> V := VectorSpace (F, d);
> o := #G;
> if #V lt #G then "Vector space too small -- no regular orbit";
> return false, true;
> end if;
>
> O := {};
> nmr := 0;
> "Order of G is ", #G;
> repeat
> repeat
> v := Random (V);
> until not (v in O);
> o := Orbit (G, v);
> if #o eq #G then "Found regular orbit"; return true, true; end if;
> O join:= o;
> "... #O is now ", #O;
> nmr +:= 1;
> decided := #V - #O lt #G;
> until decided or nmr gt Limit;
> if decided then
> "Proved no regular orbit"; return false, true;
> end if;
> if nmr gt Limit then
> "Could not decide about regular orbit"; return false, false;
> end if;
> end function;
>
> /* find space centralised by g */
> CentralisedSpace := function (g)
> G := Parent (g);
> F := CoefficientRing (G);
> A := MatrixAlgebra (F, Degree (G));
> a := A!g;
> N := NullSpace (a - Identity (A));
> // "Nullspace has dimension ", Dimension (N);
> return N;
> end function;
>
> RefinedBound := function (G, n)
> C := LMGClasses (G);
> order := #G;
> index := [i : i in [1..#C] | IsPrime (C[i][1]) and order mod C[i][1] eq 0];
> reps := [C[i][3]: i in index];
> sizes := [C[i][2]: i in index];
> spaces := [#CentralisedSpace (reps[i]) : i in [1..#reps]];
> size := &+[sizes[i] * spaces[i]: i in [1..#spaces]];
> p := Characteristic (BaseRing (G));
> e := Ilog (p, size);
> assert p^(e + 1) ge size;
> if p^e eq size then return e; else return e + 1; end if;
> end function;
>
> Scalars := function (d, q)
> F := GF (q);
> nu := PrimitiveElement (F);
> M := MatrixAlgebra (F, d);
> s := ScalarMatrix (M, nu);
> return sub;
> end function;
>
> // H defined over GF(q); construct G = H \circ scalars of F_q
> AddScalar := function (H)
> F := BaseRing (H);
> if #F gt 2 then
> S := Scalars (Degree (H), #F);
> G := sub;
> return G;
> else
> return H;
> end if;
> end function;
>
> // L list of reps; n degree; Scalar: add scalar
> // decide if rep has regular orbit
> ProcessReps := function (L, n: Scalar := true)
> for i in [1..#L] do
> "Consider the following repn", i;
> H := L[i];
> G := Scalar select AddScalar (H) else H;
>
> F := BaseRing (G);
> p := Characteristic (F);
>
> "Input degree = ", Degree (G), " Defining field size = ", #F;
>
> // replace by absolute representation
> if not IsPrime (#F) and not IsAbsolutelyIrreducible (G) then
> G := AbsoluteRepresentation (G);
> "Replaced G by its absolute representation";
> "New degree = ", Degree (G), "New field size = ", #F;
> end if;
>
> // is G conjugate to group defined over smaller field?
> flag, H := IsOverSmallerField (G);
> if flag then
> G := H;
> "Conjugated G to smaller field";
> "Degree = ", Degree (G), "New field size = ", #F;
> end if;
>
> if not IsPrime (#F) then
> G := WriteOverSmallerField (G, GF(p));
> "Rewrite G over prime field -- now degree = ", Degree (G);
> end if;
>
> o := LMGOrder (G);
> "Composition Factors of G is ";
> LMGCompositionFactors (G);
>
> if CanHaveRegularOrbit (G) then
> e := RefinedBound (G, n);
> "Refined bound on degree is ", e;
> if Degree (G) gt e then
> "Over refined degree limit -- so G has regular orbit";
> regular := true;
> else
> regular := HasRegularOrbit (G);
> // "Has regular orbit?", regular;
> end if;
> end if;
> "========================================";
> end for;
> return true;
> end function;
>
> n := 11;
> G := Sym (n);
> RandomSchreier (G);
> order := #G;
> Dims := [[32,144], [34,45], [43,45,55], [44,45], [36,44]];
> Primes := [2, 3, 5, 7, 11];
>
> for i in [1..#Primes] do
for> p := Primes[i]; dims := Dims[i];
for> "\n\nProcess n = ", n, "p = ", p;
for> I := IrreducibleModules (G, GF (p): MaxDegree := Maximum (dims));
for> J := [x : x in I | Dimension (x) in dims];
for> L := [ActionGroup (j): j in J];
for> O := [LMGOrder (l): l in L];
for> L := [L[i]: i in [1..#L] | O[i] eq #G];
for> "Faithful reps over GF(", p, ") have dimensions:", [Degree (l): l in L],
"\n";
for> f := ProcessReps (L, n);
for> end for;
Process n = 11 p = 2
Faithful reps over GF( 2 ) have dimensions: [ 32, 144 ]
Consider the following repn 1
Input degree = 32 Defining field size = 2
Composition Factors of G is
G
| Cyclic(2)
*
| Alternating(11)
1
Refined bound on degree is 32
Order of G is 39916800
Found regular orbit
========================================
Consider the following repn 2
Input degree = 144 Defining field size = 2
Composition Factors of G is
G
| Cyclic(2)
*
| Alternating(11)
1
#Warning: we will need to find a perm rep of the radical quotient!
#Found perm rep of the radical quotient!
Refined bound on degree is 89
Over refined degree limit -- so G has regular orbit
========================================
Process n = 11 p = 3
Faithful reps over GF( 3 ) have dimensions: [ 34, 34, 45, 45 ]
Consider the following repn 1
Input degree = 34 Defining field size = 3
Composition Factors of G is
G
| Cyclic(2)
*
| Alternating(11)
*
| Cyclic(2)
1
Refined bound on degree is 31
Over refined degree limit -- so G has regular orbit
========================================
Consider the following repn 2
Input degree = 34 Defining field size = 3
Composition Factors of G is
G
| Cyclic(2)
*
| Alternating(11)
*
| Cyclic(2)
1
Refined bound on degree is 31
Over refined degree limit -- so G has regular orbit
========================================
Consider the following repn 3
Input degree = 45 Defining field size = 3
Composition Factors of G is
G
| Cyclic(2)
*
| Alternating(11)
*
| Cyclic(2)
1
Refined bound on degree is 40
Over refined degree limit -- so G has regular orbit
========================================
Consider the following repn 4
Input degree = 45 Defining field size = 3
Composition Factors of G is
G
| Cyclic(2)
*
| Alternating(11)
*
| Cyclic(2)
1
Refined bound on degree is 40
Over refined degree limit -- so G has regular orbit
========================================
Process n = 11 p = 5
Faithful reps over GF( 5 ) have dimensions: [ 43, 43, 45, 45, 55, 55 ]
Consider the following repn 1
Input degree = 43 Defining field size = 5
Composition Factors of G is
G
| Cyclic(2)
*
| Alternating(11)
*
| Cyclic(2)
*
| Cyclic(2)
1
Refined bound on degree is 38
Over refined degree limit -- so G has regular orbit
========================================
Consider the following repn 2
Input degree = 43 Defining field size = 5
Composition Factors of G is
G
| Cyclic(2)
*
| Alternating(11)
*
| Cyclic(2)
*
| Cyclic(2)
1
Refined bound on degree is 38
Over refined degree limit -- so G has regular orbit
========================================
Consider the following repn 3
Input degree = 45 Defining field size = 5
Composition Factors of G is
G
| Cyclic(2)
*
| Alternating(11)
*
| Cyclic(2)
*
| Cyclic(2)
1
Refined bound on degree is 39
Over refined degree limit -- so G has regular orbit
========================================
Consider the following repn 4
Input degree = 45 Defining field size = 5
Composition Factors of G is
G
| Cyclic(2)
*
| Alternating(11)
*
| Cyclic(2)
*
| Cyclic(2)
1
Refined bound on degree is 39
Over refined degree limit -- so G has regular orbit
========================================
Consider the following repn 5
Input degree = 55 Defining field size = 5
Composition Factors of G is
G
| Cyclic(2)
*
| Alternating(11)
*
| Cyclic(2)
*
| Cyclic(2)
1
Refined bound on degree is 37
Over refined degree limit -- so G has regular orbit
========================================
Consider the following repn 6
Input degree = 55 Defining field size = 5
Composition Factors of G is
G
| Cyclic(2)
*
| Alternating(11)
*
| Cyclic(2)
*
| Cyclic(2)
1
Refined bound on degree is 37
Over refined degree limit -- so G has regular orbit
========================================
Process n = 11 p = 7
Faithful reps over GF( 7 ) have dimensions: [ 44, 44, 45, 45 ]
Consider the following repn 1
Input degree = 44 Defining field size = 7
Composition Factors of G is
G
| Cyclic(2)
*
| Alternating(11)
*
| Cyclic(3)
*
| Cyclic(2)
1
Refined bound on degree is 39
Over refined degree limit -- so G has regular orbit
========================================
Consider the following repn 2
Input degree = 44 Defining field size = 7
Composition Factors of G is
G
| Cyclic(2)
*
| Alternating(11)
*
| Cyclic(3)
*
| Cyclic(2)
1
Refined bound on degree is 39
Over refined degree limit -- so G has regular orbit
========================================
Consider the following repn 3
Input degree = 45 Defining field size = 7
Composition Factors of G is
G
| Cyclic(2)
*
| Alternating(11)
*
| Cyclic(3)
*
| Cyclic(2)
1
Refined bound on degree is 39
Over refined degree limit -- so G has regular orbit
========================================
Consider the following repn 4
Input degree = 45 Defining field size = 7
Composition Factors of G is
G
| Cyclic(2)
*
| Alternating(11)
*
| Cyclic(3)
*
| Cyclic(2)
1
Refined bound on degree is 39
Over refined degree limit -- so G has regular orbit
========================================
Process n = 11 p = 11
Faithful reps over GF( 11 ) have dimensions: [ 36, 36, 44, 44 ]
Consider the following repn 1
Input degree = 36 Defining field size = 11
Composition Factors of G is
G
| Cyclic(2)
*
| Alternating(11)
*
| Cyclic(5)
*
| Cyclic(2)
1
Refined bound on degree is 30
Over refined degree limit -- so G has regular orbit
========================================
Consider the following repn 2
Input degree = 36 Defining field size = 11
Composition Factors of G is
G
| Cyclic(2)
*
| Alternating(11)
*
| Cyclic(5)
*
| Cyclic(2)
1
Refined bound on degree is 30
Over refined degree limit -- so G has regular orbit
========================================
Consider the following repn 3
Input degree = 44 Defining field size = 11
Composition Factors of G is
G
| Cyclic(2)
*
| Alternating(11)
*
| Cyclic(5)
*
| Cyclic(2)
1
Refined bound on degree is 38
Over refined degree limit -- so G has regular orbit
========================================
Consider the following repn 4
Input degree = 44 Defining field size = 11
Composition Factors of G is
G
| Cyclic(2)
*
| Alternating(11)
*
| Cyclic(5)
*
| Cyclic(2)
1
Refined bound on degree is 38
Over refined degree limit -- so G has regular orbit
========================================
Total time: 3710.829 seconds, Total memory usage: 3027.66MB