/* Group 40.5 downloaded from the LMFDB on 08 June 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([4, -2, -2, -2, -5, 145, 21, 434, 34, 515]); a,b := Explode([GPC.1, GPC.2]); AssignNames(~GPC, ["a", "b", "b2", "b4"]);
GPerm := PermutationGroup< 9 | (2,3)(4,5)(6,7)(8,9), (6,8,7,9), (6,7)(8,9), (1,2,4,5,3) >;
GLZ := MatrixGroup< 6, Integers() | [[1, 0, 0, 0, 0, 0, -1, -1, -1, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1], [0, 0, -1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 1, 1, 1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 0]] >;
GLFp := MatrixGroup< 2, GF(5) | [[1, 1, 0, 1], [2, 0, 0, 3], [1, 0, 0, 4]] >;

 
/* Booleans */
RF := recformat< Agroup, Zgroup, abelian, almost_simple, cyclic, metabelian, metacyclic, monomial, nilpotent, perfect, quasisimple, rational, solvable, supersolvable  : BoolElt >; 
booleans_40_5 := rec< RF |  
Agroup := true,
Zgroup := false,
abelian := false,
almost_simple := false,
cyclic := false,
metabelian := true,
metacyclic := true,
monomial := true,
nilpotent := false,
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^10>,< 2, 5, a>,< 2, 5, a*b^2>,< 4, 1, b^5>,< 4, 1, b^15>,< 4, 5, a*b>,< 4, 5, a*b^3>,< 5, 2, b^4>,< 5, 2, b^8>,< 10, 2, b^2>,< 10, 2, b^6>,< 20, 2, b>,< 20, 2, b^11>,< 20, 2, b^3>,< 20, 2, b^13>]); 
CR := CharacterRing(G);
x := CR!\[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]; 
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]; 
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]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |1,-1,-1,1,-1*K.1,K.1,K.1,-1*K.1,1,1,-1,-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,K.1,-1*K.1,-1*K.1,K.1,1,1,-1,-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*K.1,K.1,-1*K.1,K.1,1,1,-1,-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,K.1,-1*K.1,K.1,-1*K.1,1,1,-1,-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(5: Sparse := true);
S := [ K |2,2,0,0,2,2,0,0,K.1^2+K.1^-2,K.1+K.1^-1,K.1+K.1^-1,K.1^2+K.1^-2,K.1^2+K.1^-2,K.1^2+K.1^-2,K.1+K.1^-1,K.1+K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |2,2,0,0,2,2,0,0,K.1+K.1^-1,K.1^2+K.1^-2,K.1^2+K.1^-2,K.1+K.1^-1,K.1+K.1^-1,K.1+K.1^-1,K.1^2+K.1^-2,K.1^2+K.1^-2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |2,2,0,0,-2,-2,0,0,K.1^2+K.1^-2,K.1+K.1^-1,K.1+K.1^-1,K.1^2+K.1^-2,-1*K.1^2-K.1^-2,-1*K.1^2-K.1^-2,-1*K.1-K.1^-1,-1*K.1-K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |2,2,0,0,-2,-2,0,0,K.1+K.1^-1,K.1^2+K.1^-2,K.1^2+K.1^-2,K.1+K.1^-1,-1*K.1-K.1^-1,-1*K.1-K.1^-1,-1*K.1^2-K.1^-2,-1*K.1^2-K.1^-2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(20: Sparse := true);
S := [ K |2,-2,0,0,-2*K.1^5,2*K.1^5,0,0,-1*K.1^2-K.1^-2,K.1^4+K.1^-4,-1*K.1^4-K.1^-4,K.1^2+K.1^-2,K.1^3+K.1^7,-1*K.1^3-K.1^7,K.1^3-K.1^5+K.1^7,-1*K.1^3+K.1^5-K.1^7]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(20: Sparse := true);
S := [ K |2,-2,0,0,2*K.1^5,-2*K.1^5,0,0,-1*K.1^2-K.1^-2,K.1^4+K.1^-4,-1*K.1^4-K.1^-4,K.1^2+K.1^-2,-1*K.1^3-K.1^7,K.1^3+K.1^7,-1*K.1^3+K.1^5-K.1^7,K.1^3-K.1^5+K.1^7]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(20: Sparse := true);
S := [ K |2,-2,0,0,-2*K.1^5,2*K.1^5,0,0,K.1^4+K.1^-4,-1*K.1^2-K.1^-2,K.1^2+K.1^-2,-1*K.1^4-K.1^-4,-1*K.1^3+K.1^5-K.1^7,K.1^3-K.1^5+K.1^7,-1*K.1^3-K.1^7,K.1^3+K.1^7]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(20: Sparse := true);
S := [ K |2,-2,0,0,2*K.1^5,-2*K.1^5,0,0,K.1^4+K.1^-4,-1*K.1^2-K.1^-2,K.1^2+K.1^-2,-1*K.1^4-K.1^-4,K.1^3-K.1^5+K.1^7,-1*K.1^3+K.1^5-K.1^7,K.1^3+K.1^7,-1*K.1^3-K.1^7]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
_ := CharacterTable(G : Check := 0); 
chartbl_40_5:= KnownIrreducibles(CR);