/* Group 432.735 downloaded from the LMFDB on 24 July 2026. */
 
/* 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, 3, 2, 2, 2, 3, 2, 3, 127, 64, 65, 135, 58, 5044, 3651, 858, 165, 16133, 5388, 1699, 1286, 124, 16470, 5501, 412, 1203]); a,b,c,d,e := Explode([GPC.1, GPC.2, GPC.3, GPC.5, GPC.6]); AssignNames(~GPC, ["a", "b", "c", "c2", "d", "e", "e2"]);
GPerm := PermutationGroup< 11 | (10,11), (2,4,5)(3,8,6), (2,5,6,8)(3,7,4,9), (2,4,6,3)(5,7,8,9), (2,6)(3,4)(5,8)(7,9), (1,2,6)(3,7,5)(4,8,9), (1,3,4)(2,7,8)(5,9,6) >;
GLFp := MatrixGroup< 3, GF(3) | [[2, 0, 0, 0, 2, 0, 0, 0, 2], [1, 1, 1, 2, 2, 0, 2, 0, 2], [2, 0, 0, 0, 1, 1, 0, 0, 2], [1, 2, 2, 1, 2, 0, 1, 0, 2], [1, 1, 2, 0, 1, 0, 0, 0, 1], [2, 1, 2, 0, 0, 2, 0, 1, 1], [1, 1, 2, 1, 2, 2, 1, 0, 1]] >;

 
/* Booleans */
RF := recformat< Agroup, Zgroup, abelian, almost_simple, cyclic, metabelian, metacyclic, monomial, nilpotent, perfect, quasisimple, rational, solvable, supersolvable  : BoolElt >; 
booleans_432_735 := rec< RF |  
Agroup := false,
Zgroup := false,
abelian := false,
almost_simple := false,
cyclic := false,
metabelian := false,
metacyclic := false,
monomial := false,
nilpotent := false,
perfect := false,
quasisimple := false,
rational := false,
solvable := true,
supersolvable := false>; 

 
/* Character Table */
G:= GLFp;
C := SequenceToConjugacyClasses([car<Integers(), Integers(), G> |< 1, 1, Matrix(3, [1, 0, 0, 0, 1, 0, 0, 0, 1])>,< 2, 1, Matrix(3, [2, 0, 0, 0, 2, 0, 0, 0, 2])>,< 2, 9, Matrix(3, [2, 1, 2, 0, 0, 2, 0, 2, 0])>,< 2, 9, Matrix(3, [1, 2, 1, 0, 1, 0, 0, 2, 2])>,< 3, 8, Matrix(3, [1, 0, 0, 0, 2, 2, 0, 1, 0])>,< 3, 12, Matrix(3, [1, 0, 1, 0, 1, 0, 0, 0, 1])>,< 3, 12, Matrix(3, [1, 0, 2, 0, 1, 0, 0, 0, 1])>,< 3, 24, Matrix(3, [1, 2, 2, 0, 0, 1, 0, 2, 2])>,< 3, 24, Matrix(3, [1, 0, 2, 0, 2, 2, 0, 1, 0])>,< 4, 54, Matrix(3, [1, 0, 2, 2, 1, 1, 2, 2, 0])>,< 4, 54, Matrix(3, [2, 1, 1, 2, 1, 0, 2, 0, 1])>,< 6, 8, Matrix(3, [2, 0, 0, 0, 0, 2, 0, 1, 1])>,< 6, 12, Matrix(3, [2, 0, 1, 0, 2, 0, 0, 0, 2])>,< 6, 12, Matrix(3, [2, 0, 2, 0, 2, 0, 0, 0, 2])>,< 6, 24, Matrix(3, [2, 0, 1, 0, 1, 1, 0, 2, 0])>,< 6, 24, Matrix(3, [2, 1, 1, 0, 0, 2, 0, 1, 1])>,< 6, 36, Matrix(3, [0, 0, 2, 1, 0, 2, 1, 1, 1])>,< 6, 36, Matrix(3, [2, 1, 0, 2, 2, 1, 2, 0, 0])>,< 6, 36, Matrix(3, [0, 1, 1, 1, 0, 1, 1, 2, 2])>,< 6, 36, Matrix(3, [1, 0, 1, 2, 2, 1, 2, 1, 2])>]); 
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]; 
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]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |1,1,1,1,1,K.1^-1,K.1,K.1^-1,K.1,1,1,1,K.1,K.1^-1,K.1,K.1^-1,K.1,K.1^-1,K.1^-1,K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |1,1,1,1,1,K.1,K.1^-1,K.1,K.1^-1,1,1,1,K.1^-1,K.1,K.1^-1,K.1,K.1^-1,K.1,K.1,K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |1,-1,1,-1,1,K.1^-1,K.1,K.1^-1,K.1,-1,1,-1,-1*K.1,-1*K.1^-1,-1*K.1,-1*K.1^-1,-1*K.1,-1*K.1^-1,K.1^-1,K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |1,-1,1,-1,1,K.1,K.1^-1,K.1,K.1^-1,-1,1,-1,-1*K.1^-1,-1*K.1,-1*K.1^-1,-1*K.1,-1*K.1^-1,-1*K.1,K.1,K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
x := CR!\[2, -2, -2, 2, 2, -1, -1, -1, -1, 0, 0, -2, 1, 1, 1, 1, -1, -1, 1, 1]; 
x`IsCharacter := true;
x`Schur := -1;
x`IsIrreducible := true; 
x := CR!\[2, 2, -2, -2, 2, -1, -1, -1, -1, 0, 0, 2, -1, -1, -1, -1, 1, 1, 1, 1]; 
x`IsCharacter := true;
x`Schur := -1;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |2,-2,-2,2,2,-1*K.1,-1*K.1^-1,-1*K.1,-1*K.1^-1,0,0,-2,K.1^-1,K.1,K.1^-1,K.1,-1*K.1^-1,-1*K.1,K.1,K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |2,-2,-2,2,2,-1*K.1^-1,-1*K.1,-1*K.1^-1,-1*K.1,0,0,-2,K.1,K.1^-1,K.1,K.1^-1,-1*K.1,-1*K.1^-1,K.1^-1,K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |2,2,-2,-2,2,-1*K.1,-1*K.1^-1,-1*K.1,-1*K.1^-1,0,0,2,-1*K.1^-1,-1*K.1,-1*K.1^-1,-1*K.1,K.1^-1,K.1,K.1,K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |2,2,-2,-2,2,-1*K.1^-1,-1*K.1,-1*K.1^-1,-1*K.1,0,0,2,-1*K.1,-1*K.1^-1,-1*K.1,-1*K.1^-1,K.1,K.1^-1,K.1^-1,K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
x := CR!\[3, 3, 3, 3, 3, 0, 0, 0, 0, -1, -1, 3, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[3, -3, 3, -3, 3, 0, 0, 0, 0, 1, -1, -3, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[8, 8, 0, 0, -1, 2, 2, -1, -1, 0, 0, -1, 2, 2, -1, -1, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[8, -8, 0, 0, -1, 2, 2, -1, -1, 0, 0, 1, -2, -2, 1, 1, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |8,8,0,0,-1,2*K.1^-1,2*K.1,-1*K.1^-1,-1*K.1,0,0,-1,2*K.1,2*K.1^-1,-1*K.1,-1*K.1^-1,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |8,8,0,0,-1,2*K.1,2*K.1^-1,-1*K.1,-1*K.1^-1,0,0,-1,2*K.1^-1,2*K.1,-1*K.1^-1,-1*K.1,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |8,-8,0,0,-1,2*K.1^-1,2*K.1,-1*K.1^-1,-1*K.1,0,0,1,-2*K.1,-2*K.1^-1,K.1,K.1^-1,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |8,-8,0,0,-1,2*K.1,2*K.1^-1,-1*K.1,-1*K.1^-1,0,0,1,-2*K.1^-1,-2*K.1,K.1^-1,K.1,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
_ := CharacterTable(G : Check := 0); 
chartbl_432_735:= KnownIrreducibles(CR);