/* Group 125.3 downloaded from the LMFDB on 13 September 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([3, -5, 5, -5, 631]); a,b,c := Explode([GPC.1, GPC.2, GPC.3]); AssignNames(~GPC, ["a", "b", "c"]);
GPerm := PermutationGroup< 25 | (1,6,12,19,22)(2,7,13,20,23)(3,8,14,16,24)(4,9,15,17,25)(5,10,11,18,21), (1,21,16,11,6)(2,22,17,12,7)(3,23,18,13,8)(4,24,19,14,9)(5,25,20,15,10), (1,5,4,3,2)(6,10,9,8,7)(11,15,14,13,12)(16,20,19,18,17)(21,25,24,23,22) >;
GLFp := MatrixGroup< 3, GF(5) | [[4, 4, 4, 1, 4, 3, 3, 4, 0], [2, 0, 4, 2, 1, 3, 1, 0, 0], [1, 3, 4, 1, 2, 2, 0, 3, 0]] >;

 
/* Booleans */
RF := recformat< Agroup, Zgroup, abelian, almost_simple, cyclic, metabelian, metacyclic, monomial, nilpotent, perfect, quasisimple, rational, solvable, supersolvable  : BoolElt >; 
booleans_125_3 := 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)>,< 5, 1, c>,< 5, 1, c^4>,< 5, 1, c^2>,< 5, 1, c^3>,< 5, 5, b>,< 5, 5, b^4>,< 5, 5, b^2>,< 5, 5, b^3>,< 5, 5, a>,< 5, 5, a^4>,< 5, 5, a^2>,< 5, 5, a^3>,< 5, 5, a*b>,< 5, 5, a^4*b^4>,< 5, 5, a^2*b^2>,< 5, 5, a^3*b^3>,< 5, 5, a*b^2>,< 5, 5, a^4*b^3>,< 5, 5, a^2*b^4>,< 5, 5, a^3*b>,< 5, 5, a^2*b>,< 5, 5, a^3*b^4>,< 5, 5, a^4*b^2>,< 5, 5, a*b^3>,< 5, 5, a*b^4>,< 5, 5, a^4*b>,< 5, 5, a^2*b^3>,< 5, 5, a^3*b^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, 1, 1, 1, 1, 1, 1, 1, 1, 1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |1,1,1,1,1,K.1^-2,K.1,K.1^-1,K.1^-1,K.1^-1,K.1^-2,K.1^2,K.1,K.1^2,K.1^2,K.1^2,K.1^-2,K.1,1,K.1,K.1^-1,K.1^-2,1,K.1^-2,1,K.1^2,1,K.1^-1,K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |1,1,1,1,1,K.1^2,K.1^-1,K.1,K.1,K.1,K.1^2,K.1^-2,K.1^-1,K.1^-2,K.1^-2,K.1^-2,K.1^2,K.1^-1,1,K.1^-1,K.1,K.1^2,1,K.1^2,1,K.1^-2,1,K.1,K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |1,1,1,1,1,K.1^-1,K.1^-2,K.1^2,K.1^2,K.1^2,K.1^-1,K.1,K.1^-2,K.1,K.1,K.1,K.1^-1,K.1^-2,1,K.1^-2,K.1^2,K.1^-1,1,K.1^-1,1,K.1,1,K.1^2,K.1^-2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |1,1,1,1,1,K.1,K.1^2,K.1^-2,K.1^-2,K.1^-2,K.1,K.1^-1,K.1^2,K.1^-1,K.1^-1,K.1^-1,K.1,K.1^2,1,K.1^2,K.1^-2,K.1,1,K.1,1,K.1^-1,1,K.1^-2,K.1^2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |1,1,1,1,1,K.1^-2,K.1,K.1^-1,K.1^-2,K.1^2,K.1^2,K.1^2,1,K.1,K.1^-2,K.1^-1,1,K.1^-2,K.1^2,K.1^-1,1,K.1^-1,K.1,K.1,K.1^-2,1,K.1^-1,K.1,K.1^2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |1,1,1,1,1,K.1^2,K.1^-1,K.1,K.1^2,K.1^-2,K.1^-2,K.1^-2,1,K.1^-1,K.1^2,K.1,1,K.1^2,K.1^-2,K.1,1,K.1,K.1^-1,K.1^-1,K.1^2,1,K.1,K.1^-1,K.1^-2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |1,1,1,1,1,K.1^-1,K.1^-2,K.1^2,K.1^-1,K.1,K.1,K.1,1,K.1^-2,K.1^-1,K.1^2,1,K.1^-1,K.1,K.1^2,1,K.1^2,K.1^-2,K.1^-2,K.1^-1,1,K.1^2,K.1^-2,K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |1,1,1,1,1,K.1,K.1^2,K.1^-2,K.1,K.1^-1,K.1^-1,K.1^-1,1,K.1^2,K.1,K.1^-2,1,K.1,K.1^-1,K.1^-2,1,K.1^-2,K.1^2,K.1^2,K.1,1,K.1^-2,K.1^2,K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |1,1,1,1,1,K.1^-2,K.1,K.1^-1,K.1,K.1^-2,1,K.1^2,K.1^-2,K.1^-1,1,K.1^-2,K.1^-1,K.1^2,K.1,1,K.1^2,K.1,K.1^-2,K.1^2,K.1^-1,K.1,K.1^2,1,K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |1,1,1,1,1,K.1^2,K.1^-1,K.1,K.1^-1,K.1^2,1,K.1^-2,K.1^2,K.1,1,K.1^2,K.1,K.1^-2,K.1^-1,1,K.1^-2,K.1^-1,K.1^2,K.1^-2,K.1,K.1^-1,K.1^-2,1,K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |1,1,1,1,1,K.1^-1,K.1^-2,K.1^2,K.1^-2,K.1^-1,1,K.1,K.1^-1,K.1^2,1,K.1^-1,K.1^2,K.1,K.1^-2,1,K.1,K.1^-2,K.1^-1,K.1,K.1^2,K.1^-2,K.1,1,K.1^2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |1,1,1,1,1,K.1,K.1^2,K.1^-2,K.1^2,K.1,1,K.1^-1,K.1,K.1^-2,1,K.1,K.1^-2,K.1^-1,K.1^2,1,K.1^-1,K.1^2,K.1,K.1^-1,K.1^-2,K.1^2,K.1^-1,1,K.1^-2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |1,1,1,1,1,K.1^-2,K.1,K.1^-1,K.1^2,1,K.1,K.1^2,K.1^-1,1,K.1^-1,K.1,K.1^2,1,K.1^-1,K.1^2,K.1,1,K.1^2,K.1^-1,K.1,K.1^-2,K.1^-2,K.1^-2,K.1^-2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |1,1,1,1,1,K.1^2,K.1^-1,K.1,K.1^-2,1,K.1^-1,K.1^-2,K.1,1,K.1,K.1^-1,K.1^-2,1,K.1,K.1^-2,K.1^-1,1,K.1^-2,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 := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |1,1,1,1,1,K.1^-1,K.1^-2,K.1^2,K.1,1,K.1^-2,K.1,K.1^2,1,K.1^2,K.1^-2,K.1,1,K.1^2,K.1,K.1^-2,1,K.1,K.1^2,K.1^-2,K.1^-1,K.1^-1,K.1^-1,K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |1,1,1,1,1,K.1,K.1^2,K.1^-2,K.1^-1,1,K.1^2,K.1^-1,K.1^-2,1,K.1^-2,K.1^2,K.1^-1,1,K.1^-2,K.1^-1,K.1^2,1,K.1^-1,K.1^-2,K.1^2,K.1,K.1,K.1,K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |1,1,1,1,1,K.1^-2,K.1,K.1^-1,1,K.1,K.1^-1,K.1^2,K.1^2,K.1^-2,K.1,1,K.1,K.1^-1,K.1^-2,K.1^-2,K.1^-2,K.1^2,K.1^-1,1,K.1^2,K.1^-1,K.1,K.1^2,1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |1,1,1,1,1,K.1^2,K.1^-1,K.1,1,K.1^-1,K.1,K.1^-2,K.1^-2,K.1^2,K.1^-1,1,K.1^-1,K.1,K.1^2,K.1^2,K.1^2,K.1^-2,K.1,1,K.1^-2,K.1,K.1^-1,K.1^-2,1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |1,1,1,1,1,K.1^-1,K.1^-2,K.1^2,1,K.1^-2,K.1^2,K.1,K.1,K.1^-1,K.1^-2,1,K.1^-2,K.1^2,K.1^-1,K.1^-1,K.1^-1,K.1,K.1^2,1,K.1,K.1^2,K.1^-2,K.1,1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |1,1,1,1,1,K.1,K.1^2,K.1^-2,1,K.1^2,K.1^-2,K.1^-1,K.1^-1,K.1,K.1^2,1,K.1^2,K.1^-2,K.1,K.1,K.1,K.1^-1,K.1^-2,1,K.1^-1,K.1^-2,K.1^2,K.1^-1,1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |1,1,1,1,1,1,1,1,K.1^-2,K.1,K.1^-2,1,K.1^-2,K.1^-2,K.1^2,K.1^-1,K.1^-1,K.1^-1,K.1^-1,K.1,K.1^2,K.1^2,K.1^2,K.1,K.1,K.1,K.1^-2,K.1^-1,K.1^2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |1,1,1,1,1,1,1,1,K.1^2,K.1^-1,K.1^2,1,K.1^2,K.1^2,K.1^-2,K.1,K.1,K.1,K.1,K.1^-1,K.1^-2,K.1^-2,K.1^-2,K.1^-1,K.1^-1,K.1^-1,K.1^2,K.1,K.1^-2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |1,1,1,1,1,1,1,1,K.1^-1,K.1^-2,K.1^-1,1,K.1^-1,K.1^-1,K.1,K.1^2,K.1^2,K.1^2,K.1^2,K.1^-2,K.1,K.1,K.1,K.1^-2,K.1^-2,K.1^-2,K.1^-1,K.1^2,K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |1,1,1,1,1,1,1,1,K.1,K.1^2,K.1,1,K.1,K.1,K.1^-1,K.1^-2,K.1^-2,K.1^-2,K.1^-2,K.1^2,K.1^-1,K.1^-1,K.1^-1,K.1^2,K.1^2,K.1^2,K.1,K.1^-2,K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |5,5*K.1^-2,5*K.1^2,5*K.1^-1,5*K.1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |5,5*K.1^2,5*K.1^-2,5*K.1,5*K.1^-1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |5,5*K.1^-1,5*K.1,5*K.1^2,5*K.1^-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |5,5*K.1,5*K.1^-1,5*K.1^-2,5*K.1^2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
_ := CharacterTable(G : Check := 0); 
chartbl_125_3:= KnownIrreducibles(CR);