/* Group 64.127 downloaded from the LMFDB on 25 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([6, 2, 2, 2, 2, 2, 2, 24, 73, 31, 489, 69, 88]); a,b,c := Explode([GPC.1, GPC.2, GPC.4]); AssignNames(~GPC, ["a", "b", "b2", "c", "c2", "c4"]);
GPerm := PermutationGroup< 16 | (1,2,5,4,3,6,8,7)(9,10,12,14)(11,15,16,13), (1,3)(2,6)(4,7)(5,8)(9,11,12,16)(10,13,14,15), (1,4,5,6,3,7,8,2), (9,12)(10,14)(11,16)(13,15), (1,5,3,8)(2,4,6,7)(9,12)(10,14)(11,16)(13,15), (1,3)(2,6)(4,7)(5,8) >;
GLFp := MatrixGroup< 4, GF(5) | [[4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 0, 4], [2, 0, 0, 0, 4, 3, 2, 3, 2, 3, 3, 4, 2, 3, 1, 1], [4, 0, 0, 0, 1, 3, 3, 3, 1, 4, 2, 1, 0, 0, 0, 1], [4, 0, 0, 0, 3, 1, 4, 0, 1, 4, 2, 3, 2, 3, 1, 1], [1, 1, 4, 4, 3, 0, 1, 1, 1, 1, 0, 4, 0, 0, 0, 1], [2, 0, 0, 0, 2, 3, 0, 0, 1, 0, 3, 1, 0, 0, 0, 2]] >;

 
/* Booleans */
RF := recformat< Agroup, Zgroup, abelian, almost_simple, cyclic, metabelian, metacyclic, monomial, nilpotent, perfect, quasisimple, rational, solvable, supersolvable  : BoolElt >; 
booleans_64_127 := 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:= GPC;
C := SequenceToConjugacyClasses([car<Integers(), Integers(), G> |< 1, 1, Id(G)>,< 2, 1, b^2>,< 2, 1, b^2*c^4>,< 2, 1, c^4>,< 4, 1, c^2>,< 4, 1, c^6>,< 4, 1, b^2*c^2>,< 4, 1, b^2*c^6>,< 4, 2, a>,< 4, 2, a*c^4>,< 4, 2, a*c^2>,< 4, 2, a*c^6>,< 4, 4, b>,< 4, 4, b*c^2>,< 4, 4, a*b>,< 4, 4, a*b*c^2>,< 8, 2, c>,< 8, 2, c^3>,< 8, 2, b^2*c>,< 8, 2, b^2*c^3>,< 8, 2, a*c>,< 8, 2, a*c^3>,< 8, 2, a*c^7>,< 8, 2, a*c^5>,< 8, 4, b*c>,< 8, 4, b*c^3>,< 8, 4, a*b*c>,< 8, 4, a*b*c^3>]); 
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]; 
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]; 
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]; 
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]; 
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]; 
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]; 
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]; 
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]; 
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*K.1,-1*K.1,K.1,-1*K.1,-1*K.1,K.1,K.1,K.1,-1*K.1,K.1,K.1,-1*K.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,K.1,K.1,-1*K.1,K.1,K.1,-1*K.1,-1*K.1,-1*K.1,K.1,-1*K.1,-1*K.1,K.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*K.1,-1*K.1,K.1,-1*K.1,-1*K.1,K.1,K.1,K.1,K.1,-1*K.1,-1*K.1,K.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,K.1,K.1,-1*K.1,K.1,K.1,-1*K.1,-1*K.1,-1*K.1,-1*K.1,K.1,K.1,-1*K.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*K.1,-1*K.1,-1*K.1,K.1,K.1,-1*K.1,K.1,K.1,K.1,-1*K.1,K.1,-1*K.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,K.1,K.1,K.1,-1*K.1,-1*K.1,K.1,-1*K.1,-1*K.1,-1*K.1,K.1,-1*K.1,K.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*K.1,-1*K.1,-1*K.1,K.1,K.1,-1*K.1,K.1,K.1,-1*K.1,K.1,-1*K.1,K.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,K.1,K.1,K.1,-1*K.1,-1*K.1,K.1,-1*K.1,-1*K.1,K.1,-1*K.1,K.1,-1*K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
x := CR!\[2, -2, -2, 2, 2, 2, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0, -2, 2, 0, 0, 0, 0, -2, 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, 0, 0, 0, 0, 0, 2, -2, 0, 0, 0, 0, 2, -2, 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*K.1,2*K.1,-2*K.1,2*K.1,-2,2,2*K.1,-2*K.1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |2,2,-2,-2,2*K.1,-2*K.1,2*K.1,-2*K.1,-2,2,-2*K.1,2*K.1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |2,2,-2,-2,-2*K.1,2*K.1,-2*K.1,2*K.1,2,-2,-2*K.1,2*K.1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |2,2,-2,-2,2*K.1,-2*K.1,2*K.1,-2*K.1,2,-2,2*K.1,-2*K.1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |2,-2,-2,2,-2,-2,2,2,0,0,0,0,0,0,0,0,-2*K.1,2*K.1,0,0,0,0,2*K.1,-2*K.1,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |2,-2,-2,2,-2,-2,2,2,0,0,0,0,0,0,0,0,2*K.1,-2*K.1,0,0,0,0,-2*K.1,2*K.1,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(8: Sparse := true);
S := [ K |2,-2,2,-2,-2*K.1^2,2*K.1^2,2*K.1^2,-2*K.1^2,0,0,0,0,0,0,0,0,0,0,-2*K.1,-2*K.1^3,2*K.1^3,2*K.1,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(8: Sparse := true);
S := [ K |2,-2,2,-2,2*K.1^2,-2*K.1^2,-2*K.1^2,2*K.1^2,0,0,0,0,0,0,0,0,0,0,2*K.1^3,2*K.1,-2*K.1,-2*K.1^3,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(8: Sparse := true);
S := [ K |2,-2,2,-2,-2*K.1^2,2*K.1^2,2*K.1^2,-2*K.1^2,0,0,0,0,0,0,0,0,0,0,2*K.1,2*K.1^3,-2*K.1^3,-2*K.1,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(8: Sparse := true);
S := [ K |2,-2,2,-2,2*K.1^2,-2*K.1^2,-2*K.1^2,2*K.1^2,0,0,0,0,0,0,0,0,0,0,-2*K.1^3,-2*K.1,2*K.1,2*K.1^3,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
_ := CharacterTable(G : Check := 0); 
chartbl_64_127:= KnownIrreducibles(CR);