/* Group 256.6076 downloaded from the LMFDB on 14 October 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([8, 2, 2, 2, 2, 2, 2, 2, 2, 66, 1780, 268, 116, 3702, 1142, 1374, 166, 2583]); a,b,c,d,e := Explode([GPC.1, GPC.2, GPC.3, GPC.5, GPC.7]); AssignNames(~GPC, ["a", "b", "c", "c2", "d", "d2", "e", "e2"]);
GPerm := PermutationGroup< 12 | (1,2,5,6)(3,7,8,4)(9,10,11,12), (1,3)(2,4)(5,6,7,8)(10,12), (1,4)(2,3)(5,7)(6,8)(10,12), (1,5,4,7)(2,6,3,8)(9,11)(10,12), (9,11)(10,12), (1,5)(2,6)(3,8)(4,7)(9,11)(10,12), (1,4)(5,7), (1,4)(2,3)(5,7)(6,8) >;
GLZ := MatrixGroup< 6, Integers() | [[0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0], [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, 1, 0, 0, 0, 0, 0, 0, -1], [0, 0, -1, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0]] >;

 
/* Booleans */
RF := recformat< Agroup, Zgroup, abelian, almost_simple, cyclic, metabelian, metacyclic, monomial, nilpotent, perfect, quasisimple, rational, solvable, supersolvable  : BoolElt >; 
booleans_256_6076 := 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:= GPerm;
C := SequenceToConjugacyClasses([car<Integers(), Integers(), G> |< 1, 1, Id(G)>,< 2, 1, G!(1,4)(2,3)(5,7)(6,8)(9,11)(10,12)>,< 2, 1, G!(1,4)(2,3)(5,7)(6,8)>,< 2, 1, G!(9,11)(10,12)>,< 2, 2, G!(10,12)>,< 2, 2, G!(1,4)(2,3)(5,7)(6,8)(10,12)>,< 2, 2, G!(2,3)(6,8)>,< 2, 2, G!(2,3)(6,8)(9,11)(10,12)>,< 2, 4, G!(5,7)(6,8)(10,12)>,< 2, 4, G!(5,7)(6,8)(9,11)>,< 2, 4, G!(5,7)(6,8)(9,11)(10,12)>,< 2, 4, G!(2,3)(6,8)(10,12)>,< 2, 4, G!(5,7)(6,8)>,< 2, 4, G!(1,5)(2,6)(3,8)(4,7)>,< 2, 4, G!(1,5)(2,6)(3,8)(4,7)(9,11)(10,12)>,< 2, 8, G!(1,5)(2,6)(3,8)(4,7)(10,12)>,< 2, 16, G!(2,6)(3,8)(5,7)(9,10)(11,12)>,< 4, 4, G!(1,5,4,7)(2,6,3,8)>,< 4, 4, G!(1,5,4,7)(2,6,3,8)(9,11)(10,12)>,< 4, 8, G!(2,6,3,8)(9,10)(11,12)>,< 4, 8, G!(2,6,3,8)(9,10,11,12)>,< 4, 8, G!(1,4)(2,6,3,8)(5,7)(9,10)(11,12)>,< 4, 8, G!(1,4)(2,6,3,8)(5,7)(9,10,11,12)>,< 4, 8, G!(1,5,4,7)(2,6,3,8)(10,12)>,< 4, 8, G!(1,2)(3,4)(5,6,7,8)>,< 4, 8, G!(1,2)(3,4)(5,8,7,6)>,< 4, 8, G!(1,2)(3,4)(5,6,7,8)(10,12)>,< 4, 8, G!(1,2)(3,4)(5,8,7,6)(10,12)>,< 4, 8, G!(1,2)(3,4)(5,6,7,8)(9,11)>,< 4, 8, G!(1,2)(3,4)(5,8,7,6)(9,11)>,< 4, 8, G!(1,2)(3,4)(5,6,7,8)(9,11)(10,12)>,< 4, 8, G!(1,2)(3,4)(5,8,7,6)(9,11)(10,12)>,< 4, 16, G!(2,6)(3,8)(5,7)(9,10,11,12)>,< 4, 16, G!(1,2,5,6)(3,7,8,4)(9,10)(11,12)>,< 4, 16, G!(1,2,5,8)(3,7,6,4)(9,10)(11,12)>,< 4, 16, G!(1,2,5,6)(3,7,8,4)(9,10,11,12)>,< 4, 16, G!(1,2,5,8)(3,7,6,4)(9,10,11,12)>]); 
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, 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, 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, -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, 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, -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, -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, 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, -1, -1, -1, 1, 1, 1, -1, -1, -1, -1, 1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |1,1,1,1,-1,-1,1,1,1,1,-1,-1,-1,-1,-1,1,-1,1,1,-1,-1*K.1,1,K.1,-1*K.1,K.1,K.1,-1*K.1,-1*K.1,K.1,1,-1,-1,K.1,-1*K.1,-1*K.1,K.1,1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |1,1,1,1,-1,-1,1,1,1,1,-1,-1,-1,-1,-1,1,-1,1,1,-1,K.1,1,-1*K.1,K.1,-1*K.1,-1*K.1,K.1,K.1,-1*K.1,1,-1,-1,-1*K.1,K.1,K.1,-1*K.1,1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |1,1,1,1,-1,-1,1,1,1,1,-1,-1,-1,-1,-1,1,1,1,1,1,-1*K.1,-1,K.1,-1*K.1,K.1,K.1,-1*K.1,-1*K.1,K.1,-1,1,-1,-1*K.1,K.1,K.1,-1*K.1,-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |1,1,1,1,-1,-1,1,1,1,1,-1,-1,-1,-1,-1,1,1,1,1,1,K.1,-1,-1*K.1,K.1,-1*K.1,-1*K.1,K.1,K.1,-1*K.1,-1,1,-1,K.1,-1*K.1,-1*K.1,K.1,-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |1,1,1,1,1,1,1,1,-1,-1,-1,1,-1,-1,-1,-1,-1,1,1,1,-1*K.1,1,-1*K.1,-1*K.1,K.1,-1*K.1,K.1,K.1,K.1,1,1,1,K.1,-1*K.1,K.1,-1*K.1,-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |1,1,1,1,1,1,1,1,-1,-1,-1,1,-1,-1,-1,-1,-1,1,1,1,K.1,1,K.1,K.1,-1*K.1,K.1,-1*K.1,-1*K.1,-1*K.1,1,1,1,-1*K.1,K.1,-1*K.1,K.1,-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |1,1,1,1,1,1,1,1,-1,-1,-1,1,-1,-1,-1,-1,1,1,1,-1,-1*K.1,-1,-1*K.1,-1*K.1,K.1,-1*K.1,K.1,K.1,K.1,-1,-1,1,-1*K.1,K.1,-1*K.1,K.1,1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |1,1,1,1,1,1,1,1,-1,-1,-1,1,-1,-1,-1,-1,1,1,1,-1,K.1,-1,K.1,K.1,-1*K.1,K.1,-1*K.1,-1*K.1,-1*K.1,-1,-1,1,K.1,-1*K.1,K.1,-1*K.1,1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
x := CR!\[2, -2, -2, 2, 0, 0, -2, 2, 0, 0, -2, 0, -2, 2, 2, 0, 0, 2, -2, 0, -2, 0, 0, 2, -2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, -2, -2, 2, 0, 0, -2, 2, 0, 0, -2, 0, -2, 2, 2, 0, 0, 2, -2, 0, 2, 0, 0, -2, 2, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, -2, -2, 2, 0, 0, -2, 2, 0, 0, -2, 0, 2, -2, 2, 0, 0, -2, 2, 0, 0, 0, -2, 0, 0, 2, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, -2, -2, 2, 0, 0, -2, 2, 0, 0, -2, 0, 2, -2, 2, 0, 0, -2, 2, 0, 0, 0, 2, 0, 0, -2, -2, 2, 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, -2, -2, 2, -2, -2, -2, 2, 2, 0, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 2, 2, 2, -2, -2, 2, 2, 2, 2, -2, -2, 2, 2, -2, -2, 0, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 2, 2, 2, 2, 2, 2, 2, -2, -2, -2, 2, 2, 2, -2, 2, 0, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, -2, -2, 2, -2, 0, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |2,-2,-2,2,0,0,-2,2,0,0,2,0,-2,2,-2,0,0,-2,2,0,0,0,-2*K.1,0,0,2*K.1,-2*K.1,2*K.1,0,0,0,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |2,-2,-2,2,0,0,-2,2,0,0,2,0,-2,2,-2,0,0,-2,2,0,0,0,2*K.1,0,0,-2*K.1,2*K.1,-2*K.1,0,0,0,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |2,-2,-2,2,0,0,-2,2,0,0,2,0,2,-2,-2,0,0,2,-2,0,-2*K.1,0,0,2*K.1,2*K.1,0,0,0,-2*K.1,0,0,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |2,-2,-2,2,0,0,-2,2,0,0,2,0,2,-2,-2,0,0,2,-2,0,2*K.1,0,0,-2*K.1,-2*K.1,0,0,0,2*K.1,0,0,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
x := CR!\[4, -4, -4, 4, 0, 0, 4, -4, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[4, -4, -4, 4, 0, 0, 4, -4, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[4, 4, 4, 4, -4, -4, -4, -4, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[4, 4, 4, 4, 4, 4, -4, -4, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[4, 4, -4, -4, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 2, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[4, 4, -4, -4, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 2, -2, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[4, 4, -4, -4, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[4, 4, -4, -4, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, -2, -2, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[8, -8, 8, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
_ := CharacterTable(G : Check := 0); 
chartbl_256_6076:= KnownIrreducibles(CR);