/* Group 128.734 downloaded from the LMFDB on 29 September 2025. */
 
/* Various presentations of this group are stored in this file: 
	 GPC is polycyclic presentation GPerm is permutation group 
	 GLZ, GLFp, GLZA, GLZq, GLFq if they exist are matrix groups 
 
 Many characteristics of the group are stored as booleans in a record: 
	 Agroup, Zgroup, abelian, almost_simple,cyclic, metabelian, 
	 metacyclic, monomial, nilpotent, perfect, quasisimple, rational, 
	 solvable, supersolvable 
 
 The character table is stored as chartbl_n_i where n is the order of 
 the group and i is which group of that order it is. Conjugacy classes 
 are stored in the variable 'C' with elements from the group 'G'. 
*/
 
/* Constructions */
GPC := PCGroup([7, 2, 2, 2, 2, 2, 2, 2, 85, 36, 675, 794, 80, 1411, 4037, 124]); a,b,c,d := Explode([GPC.1, GPC.2, GPC.4, GPC.6]); AssignNames(~GPC, ["a", "b", "b2", "c", "c2", "d", "d2"]);
GPerm := PermutationGroup< 12 | (1,2)(3,5)(4,7)(6,8)(9,10)(11,12), (2,5)(3,8)(6,7)(9,11)(10,12), (3,8)(5,6)(10,11), (1,3,4,8)(2,6,7,5), (9,12)(10,11), (2,7)(5,6)(9,12)(10,11), (1,4)(2,7)(3,8)(5,6) >;
GLZ := MatrixGroup< 6, Integers() | [[1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1], [1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0], [0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1]] >;
GLZN := MatrixGroup< 2, Integers(48) | [[1, 12, 0, 1], [31, 18, 0, 1], [7, 0, 24, 7], [1, 24, 0, 1], [11, 12, 1, 1], [25, 24, 24, 25], [13, 36, 36, 37]] >;
GLZq := MatrixGroup< 2, Integers(16) | [[1, 8, 0, 1], [13, 12, 4, 5], [3, 8, 12, 11], [3, 4, 1, 1], [1, 4, 0, 1], [11, 2, 1, 9], [9, 8, 8, 9]] >;

 
/* Booleans */
RF := recformat< Agroup, Zgroup, abelian, almost_simple, cyclic, metabelian, metacyclic, monomial, nilpotent, perfect, quasisimple, rational, solvable, supersolvable  : BoolElt >; 
booleans_128_734 := rec< RF |  
Agroup := false,
Zgroup := false,
abelian := false,
almost_simple := false,
cyclic := false,
metabelian := true,
metacyclic := false,
monomial := true,
nilpotent := true,
perfect := false,
quasisimple := false,
rational := false,
solvable := true,
supersolvable := true>; 

 
/* Character Table */
G:= GLZq;
C := SequenceToConjugacyClasses([car<Integers(), Integers(), G> |< 1, 1, Matrix(2, [1, 0, 0, 1])>,< 2, 1, Matrix(2, [9, 0, 8, 9])>,< 2, 1, Matrix(2, [7, 0, 8, 7])>,< 2, 1, Matrix(2, [15, 0, 0, 15])>,< 2, 2, Matrix(2, [15, 8, 0, 15])>,< 2, 2, Matrix(2, [9, 8, 8, 9])>,< 2, 8, Matrix(2, [13, 10, 4, 3])>,< 2, 8, Matrix(2, [9, 10, 8, 7])>,< 2, 8, Matrix(2, [15, 10, 0, 1])>,< 2, 8, Matrix(2, [5, 8, 15, 11])>,< 2, 8, Matrix(2, [3, 10, 12, 13])>,< 4, 2, Matrix(2, [5, 8, 12, 13])>,< 4, 2, Matrix(2, [11, 0, 4, 3])>,< 4, 2, Matrix(2, [3, 8, 12, 11])>,< 4, 2, Matrix(2, [5, 0, 12, 13])>,< 4, 4, Matrix(2, [3, 12, 12, 11])>,< 4, 4, Matrix(2, [11, 12, 4, 3])>,< 4, 4, Matrix(2, [5, 4, 12, 13])>,< 4, 4, Matrix(2, [13, 4, 4, 5])>,< 4, 8, Matrix(2, [7, 6, 1, 1])>,< 4, 8, Matrix(2, [11, 6, 1, 5])>,< 4, 8, Matrix(2, [13, 0, 15, 11])>,< 8, 8, Matrix(2, [13, 6, 15, 7])>,< 8, 8, Matrix(2, [9, 6, 15, 3])>,< 8, 8, Matrix(2, [15, 4, 1, 13])>,< 8, 8, Matrix(2, [7, 12, 1, 13])>]); 
CR := CharacterRing(G);
x := CR!\[1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, -1, -1, 1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[1, 1, 1, 1, 1, 1, -1, -1, -1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, 1, -1, 1, 1, -1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[1, 1, 1, 1, 1, 1, -1, -1, 1, -1, 1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, -1, -1, 1, 1, -1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[1, 1, 1, 1, 1, 1, -1, -1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, 1, 1, -1, -1, 1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1, 1, 1, 1, 1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[1, 1, 1, 1, 1, 1, 1, 1, -1, 1, -1, 1, 1, 1, 1, -1, -1, -1, -1, 1, 1, 1, -1, -1, -1, -1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[1, 1, 1, 1, 1, 1, 1, 1, 1, -1, 1, 1, 1, 1, 1, 1, 1, 1, 1, -1, -1, -1, -1, -1, -1, -1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, -2, 2, -2, -2, 2, 0, 0, -2, 0, 2, -2, 2, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, -2, 2, -2, -2, 2, 0, 0, 0, 0, 0, 2, -2, -2, 2, -2, -2, 2, 2, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, -2, 2, -2, -2, 2, 0, 0, 0, 0, 0, 2, -2, -2, 2, 2, 2, -2, -2, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, -2, 2, -2, -2, 2, 0, 0, 2, 0, -2, -2, 2, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, -2, 2, -2, 2, -2, 0, 0, 0, 0, 0, -2, -2, 2, 2, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 2]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, -2, 2, -2, 2, -2, 0, 0, 0, 0, 0, -2, -2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, -2]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 2, 2, 2, -2, -2, 0, 0, 0, -2, 0, 2, -2, 2, -2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 2, 2, 2, -2, -2, 0, 0, 0, 0, 0, -2, 2, -2, 2, 0, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 2, 2, 2, -2, -2, 0, 0, 0, 0, 0, -2, 2, -2, 2, 0, 0, 0, 0, 2, -2, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 2, 2, 2, -2, -2, 0, 0, 0, 2, 0, 2, -2, 2, -2, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 2, 2, 2, 2, 2, -2, 2, 0, 0, 0, -2, -2, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 2, 2, 2, 2, 2, 2, -2, 0, 0, 0, -2, -2, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |2,-2,2,-2,2,-2,0,0,0,0,0,2,2,-2,-2,0,0,0,0,0,0,0,0,-2*K.1,2*K.1,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |2,-2,2,-2,2,-2,0,0,0,0,0,2,2,-2,-2,0,0,0,0,0,0,0,0,2*K.1,-2*K.1,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
x := CR!\[4, -4, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, -2, 2, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[4, -4, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2, 2, -2, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[4, 4, -4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 2, -2, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[4, 4, -4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2, -2, 2, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
_ := CharacterTable(G : Check := 0); 
chartbl_128_734:= KnownIrreducibles(CR);