/* Group 360.119 downloaded from the LMFDB on 13 November 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< 8 | (1,3)(2,5,4)(6,7,8), (1,4,2) >;
GLZ := MatrixGroup< 6, Integers() | [[1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1], [0, 0, 0, -1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, -1]] >;
F:=GF(4); al:=F.1; GLFq := MatrixGroup< 4, F | [[0, 0, al^1, al^1], [al^1, al^2, 0, al^1], [1, al^2, al^1, 1], [al^2, al^2, al^1, 1]],[[al^2, al^2, 0, al^2], [al^2, al^2, al^2, 1], [al^2, 0, al^2, 1], [0, 0, al^2, al^2]],[[al^1, 0, 0, 0], [0, al^1, 0, 0], [0, 0, al^1, 0], [0, 0, 0, al^1]] >;

 
/* Booleans */
RF := recformat< Agroup, Zgroup, abelian, almost_simple, cyclic, metabelian, metacyclic, monomial, nilpotent, perfect, quasisimple, rational, solvable, supersolvable  : BoolElt >; 
booleans_360_119 := 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, 10, G!(4,5)>,< 2, 15, G!(1,3)(4,5)>,< 3, 1, G!(6,8,7)>,< 3, 1, G!(6,7,8)>,< 3, 20, G!(1,2,5)>,< 3, 20, G!(3,4,5)(6,7,8)>,< 3, 20, G!(3,5,4)(6,8,7)>,< 4, 30, G!(1,4,3,5)>,< 5, 24, G!(1,2,4,5,3)>,< 6, 10, G!(4,5)(6,7,8)>,< 6, 10, G!(4,5)(6,8,7)>,< 6, 15, G!(1,3)(4,5)(6,7,8)>,< 6, 15, G!(1,3)(4,5)(6,8,7)>,< 6, 20, G!(1,5,2)(3,4)>,< 6, 20, G!(1,2)(3,5,4)(6,8,7)>,< 6, 20, G!(1,2)(3,4,5)(6,7,8)>,< 12, 30, G!(1,5,3,4)(6,8,7)>,< 12, 30, G!(1,4,3,5)(6,7,8)>,< 15, 24, G!(1,4,3,2,5)(6,7,8)>,< 15, 24, G!(1,5,2,3,4)(6,8,7)>]); 
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]; 
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]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |1,1,1,K.1^-1,K.1,1,K.1,K.1^-1,1,1,K.1,K.1^-1,K.1,K.1^-1,1,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,K.1,K.1^-1,1,K.1^-1,K.1,1,1,K.1^-1,K.1,K.1^-1,K.1,1,K.1,K.1^-1,K.1,K.1^-1,K.1^-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,K.1^-1,K.1,1,K.1,K.1^-1,-1,1,-1*K.1,-1*K.1^-1,K.1,K.1^-1,-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; 
K := CyclotomicField(3: Sparse := true);
S := [ K |1,-1,1,K.1,K.1^-1,1,K.1^-1,K.1,-1,1,-1*K.1^-1,-1*K.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; 
x := CR!\[4, 2, 0, 4, 4, 1, 1, 1, 0, -1, 2, 2, 0, 0, -1, -1, -1, 0, 0, -1, -1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[4, -2, 0, 4, 4, 1, 1, 1, 0, -1, -2, -2, 0, 0, 1, 1, 1, 0, 0, -1, -1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |4,2,0,4*K.1^-1,4*K.1,1,K.1,K.1^-1,0,-1,2*K.1,2*K.1^-1,0,0,-1,-1*K.1^-1,-1*K.1,0,0,-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 |4,2,0,4*K.1,4*K.1^-1,1,K.1^-1,K.1,0,-1,2*K.1^-1,2*K.1,0,0,-1,-1*K.1,-1*K.1^-1,0,0,-1*K.1^-1,-1*K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |4,-2,0,4*K.1^-1,4*K.1,1,K.1,K.1^-1,0,-1,-2*K.1,-2*K.1^-1,0,0,1,K.1^-1,K.1,0,0,-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 |4,-2,0,4*K.1,4*K.1^-1,1,K.1^-1,K.1,0,-1,-2*K.1^-1,-2*K.1,0,0,1,K.1,K.1^-1,0,0,-1*K.1^-1,-1*K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
x := CR!\[5, -1, 1, 5, 5, -1, -1, -1, 1, 0, -1, -1, 1, 1, -1, -1, -1, 1, 1, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[5, 1, 1, 5, 5, -1, -1, -1, -1, 0, 1, 1, 1, 1, 1, 1, 1, -1, -1, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |5,-1,1,5*K.1^-1,5*K.1,-1,-1*K.1,-1*K.1^-1,1,0,-1*K.1,-1*K.1^-1,K.1,K.1^-1,-1,-1*K.1^-1,-1*K.1,K.1^-1,K.1,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |5,-1,1,5*K.1,5*K.1^-1,-1,-1*K.1^-1,-1*K.1,1,0,-1*K.1^-1,-1*K.1,K.1^-1,K.1,-1,-1*K.1,-1*K.1^-1,K.1,K.1^-1,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |5,1,1,5*K.1^-1,5*K.1,-1,-1*K.1,-1*K.1^-1,-1,0,K.1,K.1^-1,K.1,K.1^-1,1,K.1^-1,K.1,-1*K.1^-1,-1*K.1,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |5,1,1,5*K.1,5*K.1^-1,-1,-1*K.1^-1,-1*K.1,-1,0,K.1^-1,K.1,K.1^-1,K.1,1,K.1,K.1^-1,-1*K.1,-1*K.1^-1,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
x := CR!\[6, 0, -2, 6, 6, 0, 0, 0, 0, 1, 0, 0, -2, -2, 0, 0, 0, 0, 0, 1, 1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |6,0,-2,6*K.1^-1,6*K.1,0,0,0,0,1,0,0,-2*K.1,-2*K.1^-1,0,0,0,0,0,K.1,K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |6,0,-2,6*K.1,6*K.1^-1,0,0,0,0,1,0,0,-2*K.1^-1,-2*K.1,0,0,0,0,0,K.1^-1,K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
_ := CharacterTable(G : Check := 0); 
chartbl_360_119:= KnownIrreducibles(CR);