/* Group 9072.b downloaded from the LMFDB on 28 December 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 */
GPerm := PermutationGroup< 12 | (1,2)(3,5)(4,7)(6,8)(10,11), (1,3,6,9,7,5,8,2,4)(11,12) >;

 
/* Booleans */
RF := recformat< Agroup, Zgroup, abelian, almost_simple, cyclic, metabelian, metacyclic, monomial, nilpotent, perfect, quasisimple, rational, solvable, supersolvable  : BoolElt >; 
booleans_9072_b := 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 := false,
supersolvable := false>; 

 
/* Character Table */
G:= GPerm;
C := SequenceToConjugacyClasses([car<Integers(), Integers(), G> |< 1, 1, Id(G)>,< 2, 3, G!(10,12)>,< 2, 63, G!(1,2)(3,4)(6,9)(7,8)>,< 2, 189, G!(1,7)(2,9)(3,5)(4,8)(10,12)>,< 3, 2, G!(10,12,11)>,< 3, 56, G!(1,9,8)(2,3,7)(4,6,5)>,< 3, 84, G!(1,5,2)(3,9,7)>,< 3, 84, G!(1,2,5)(3,7,9)>,< 3, 112, G!(1,8,5)(2,4,3)(6,9,7)(10,12,11)>,< 3, 168, G!(1,9,7)(2,6,8)(10,12,11)>,< 3, 168, G!(1,7,9)(2,8,6)(10,11,12)>,< 6, 126, G!(1,6)(2,9)(3,8)(4,7)(10,11,12)>,< 6, 168, G!(1,6,5)(2,8,7)(3,9,4)(10,11)>,< 6, 252, G!(1,9,2)(3,4,8)(11,12)>,< 6, 252, G!(1,2,9)(3,8,4)(11,12)>,< 6, 252, G!(1,5,2,7,8,4)(3,6)>,< 6, 252, G!(1,4,8,7,2,5)(3,6)>,< 6, 504, G!(1,8,9,2,7,6)(3,4)(10,11,12)>,< 6, 504, G!(1,6,7,2,9,8)(3,4)(10,12,11)>,< 6, 756, G!(1,9,5,7,2,3)(4,8)(10,12)>,< 6, 756, G!(1,3,2,7,5,9)(4,8)(10,12)>,< 7, 216, G!(1,2,3,8,9,6,5)>,< 9, 168, G!(1,3,7,5,4,8,6,9,2)>,< 9, 168, G!(1,7,6,5,4,8,9,2,3)>,< 9, 168, G!(1,3,2,9,8,4,5,6,7)>,< 9, 336, G!(1,3,4,2,9,8,5,7,6)(10,12,11)>,< 9, 336, G!(1,7,4,9,2,6,8,3,5)(10,12,11)>,< 9, 336, G!(1,5,3,8,6,2,9,4,7)(10,11,12)>,< 14, 648, G!(1,9,2,6,3,5,8)(10,12)>,< 18, 504, G!(1,8,3,6,7,9,5,2,4)(10,11)>,< 18, 504, G!(1,8,7,9,6,2,5,3,4)(10,12)>,< 18, 504, G!(1,4,3,5,2,6,9,7,8)(10,12)>,< 21, 432, G!(2,8,3,6,5,4,9)(10,12,11)>]); 
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]; 
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]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |1,1,1,1,1,1,K.1^-1,K.1,1,K.1^-1,K.1,1,1,K.1^-1,K.1,K.1^-1,K.1,K.1,K.1^-1,K.1,K.1^-1,1,1,K.1^-1,K.1,1,K.1^-1,K.1,1,1,K.1,K.1^-1,1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |1,1,1,1,1,1,K.1,K.1^-1,1,K.1,K.1^-1,1,1,K.1,K.1^-1,K.1,K.1^-1,K.1^-1,K.1,K.1^-1,K.1,1,1,K.1,K.1^-1,1,K.1,K.1^-1,1,1,K.1^-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,-1,1,1,K.1^-1,K.1,1,K.1^-1,K.1,1,-1,-1*K.1^-1,-1*K.1,K.1^-1,K.1,K.1,K.1^-1,-1*K.1,-1*K.1^-1,1,1,K.1^-1,K.1,1,K.1^-1,K.1,-1,-1,-1*K.1,-1*K.1^-1,1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |1,-1,1,-1,1,1,K.1,K.1^-1,1,K.1,K.1^-1,1,-1,-1*K.1,-1*K.1^-1,K.1,K.1^-1,K.1^-1,K.1,-1*K.1^-1,-1*K.1,1,1,K.1,K.1^-1,1,K.1,K.1^-1,-1,-1,-1*K.1^-1,-1*K.1,1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
x := CR!\[2, 0, 2, 0, -1, 2, 2, 2, -1, -1, -1, -1, 0, 0, 0, 2, 2, -1, -1, 0, 0, 2, 2, 2, 2, -1, -1, -1, 0, 0, 0, 0, -1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |2,0,2,0,-1,2,2*K.1^-1,2*K.1,-1,-1*K.1^-1,-1*K.1,-1,0,0,0,2*K.1^-1,2*K.1,-1*K.1,-1*K.1^-1,0,0,2,2,2*K.1^-1,2*K.1,-1,-1*K.1^-1,-1*K.1,0,0,0,0,-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |2,0,2,0,-1,2,2*K.1,2*K.1^-1,-1,-1*K.1,-1*K.1^-1,-1,0,0,0,2*K.1,2*K.1^-1,-1*K.1^-1,-1*K.1,0,0,2,2,2*K.1,2*K.1^-1,-1,-1*K.1,-1*K.1^-1,0,0,0,0,-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
x := CR!\[7, -7, -1, 1, 7, -2, 1, 1, -2, 1, 1, -1, 2, -1, -1, -1, -1, -1, -1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 0, -1, -1, -1, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[7, 7, -1, -1, 7, -2, 1, 1, -2, 1, 1, -1, -2, 1, 1, -1, -1, -1, -1, -1, -1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |7,-7,-1,1,7,-2,K.1^-1,K.1,-2,K.1^-1,K.1,-1,2,-1*K.1^-1,-1*K.1,-1*K.1^-1,-1*K.1,-1*K.1,-1*K.1^-1,K.1,K.1^-1,0,1,K.1^-1,K.1,1,K.1^-1,K.1,0,-1,-1*K.1,-1*K.1^-1,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |7,-7,-1,1,7,-2,K.1,K.1^-1,-2,K.1,K.1^-1,-1,2,-1*K.1,-1*K.1^-1,-1*K.1,-1*K.1^-1,-1*K.1^-1,-1*K.1,K.1^-1,K.1,0,1,K.1,K.1^-1,1,K.1,K.1^-1,0,-1,-1*K.1^-1,-1*K.1,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |7,7,-1,-1,7,-2,K.1^-1,K.1,-2,K.1^-1,K.1,-1,-2,K.1^-1,K.1,-1*K.1^-1,-1*K.1,-1*K.1,-1*K.1^-1,-1*K.1,-1*K.1^-1,0,1,K.1^-1,K.1,1,K.1^-1,K.1,0,1,K.1,K.1^-1,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |7,7,-1,-1,7,-2,K.1,K.1^-1,-2,K.1,K.1^-1,-1,-2,K.1,K.1^-1,-1*K.1,-1*K.1^-1,-1*K.1^-1,-1*K.1,-1*K.1^-1,-1*K.1,0,1,K.1,K.1^-1,1,K.1,K.1^-1,0,1,K.1^-1,K.1,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
x := CR!\[8, 8, 0, 0, 8, -1, 2, 2, -1, 2, 2, 0, -1, 2, 2, 0, 0, 0, 0, 0, 0, 1, -1, -1, -1, -1, -1, -1, 1, -1, -1, -1, 1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[8, -8, 0, 0, 8, -1, 2, 2, -1, 2, 2, 0, 1, -2, -2, 0, 0, 0, 0, 0, 0, 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 |8,8,0,0,8,-1,2*K.1^-1,2*K.1,-1,2*K.1^-1,2*K.1,0,-1,2*K.1^-1,2*K.1,0,0,0,0,0,0,1,-1,-1*K.1^-1,-1*K.1,-1,-1*K.1^-1,-1*K.1,1,-1,-1*K.1,-1*K.1^-1,1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |8,8,0,0,8,-1,2*K.1,2*K.1^-1,-1,2*K.1,2*K.1^-1,0,-1,2*K.1,2*K.1^-1,0,0,0,0,0,0,1,-1,-1*K.1,-1*K.1^-1,-1,-1*K.1,-1*K.1^-1,1,-1,-1*K.1^-1,-1*K.1,1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |8,-8,0,0,8,-1,2*K.1^-1,2*K.1,-1,2*K.1^-1,2*K.1,0,1,-2*K.1^-1,-2*K.1,0,0,0,0,0,0,1,-1,-1*K.1^-1,-1*K.1,-1,-1*K.1^-1,-1*K.1,-1,1,K.1,K.1^-1,1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |8,-8,0,0,8,-1,2*K.1,2*K.1^-1,-1,2*K.1,2*K.1^-1,0,1,-2*K.1,-2*K.1^-1,0,0,0,0,0,0,1,-1,-1*K.1,-1*K.1^-1,-1,-1*K.1,-1*K.1^-1,-1,1,K.1^-1,K.1,1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
x := CR!\[14, 0, -2, 0, -7, -4, 2, 2, 2, -1, -1, 1, 0, 0, 0, -2, -2, 1, 1, 0, 0, 0, 2, 2, 2, -1, -1, -1, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |14,0,-2,0,-7,-4,2*K.1^-1,2*K.1,2,-1*K.1^-1,-1*K.1,1,0,0,0,-2*K.1^-1,-2*K.1,K.1,K.1^-1,0,0,0,2,2*K.1^-1,2*K.1,-1,-1*K.1^-1,-1*K.1,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |14,0,-2,0,-7,-4,2*K.1,2*K.1^-1,2,-1*K.1,-1*K.1^-1,1,0,0,0,-2*K.1,-2*K.1^-1,K.1^-1,K.1,0,0,0,2,2*K.1,2*K.1^-1,-1,-1*K.1,-1*K.1^-1,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
x := CR!\[16, 0, 0, 0, -8, -2, 4, 4, 1, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2, -2, -2, 1, 1, 1, 0, 0, 0, 0, -1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |16,0,0,0,-8,-2,4*K.1^-1,4*K.1,1,-2*K.1^-1,-2*K.1,0,0,0,0,0,0,0,0,0,0,2,-2,-2*K.1^-1,-2*K.1,1,K.1^-1,K.1,0,0,0,0,-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |16,0,0,0,-8,-2,4*K.1,4*K.1^-1,1,-2*K.1,-2*K.1^-1,0,0,0,0,0,0,0,0,0,0,2,-2,-2*K.1,-2*K.1^-1,1,K.1,K.1^-1,0,0,0,0,-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
x := CR!\[21, -21, -3, 3, 21, 3, 0, 0, 3, 0, 0, -3, -3, 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!\[21, 21, -3, -3, 21, 3, 0, 0, 3, 0, 0, -3, 3, 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!\[27, 27, 3, 3, 27, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, -1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[27, -27, 3, -3, 27, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[42, 0, -6, 0, -21, 6, 0, 0, -3, 0, 0, 3, 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!\[54, 0, 6, 0, -27, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
_ := CharacterTable(G : Check := 0); 
chartbl_9072_b:= KnownIrreducibles(CR);