/* Group 500.27 downloaded from the LMFDB on 04 May 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([5, -2, -2, -5, -5, -5, 181, 26, 242, 1808, 1613, 10004, 5009]); a,b,c,d := Explode([GPC.1, GPC.2, GPC.4, GPC.5]); AssignNames(~GPC, ["a", "b", "b2", "c", "d"]);
GPerm := PermutationGroup< 25 | (2,5)(3,4)(6,21)(7,25)(8,24)(9,23)(10,22)(11,16)(12,20)(13,19)(14,18)(15,17), (2,5)(3,4)(7,10)(8,9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24), (6,7,8,9,10)(11,13,15,12,14)(16,19,17,20,18)(21,25,24,23,22), (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], [4, 4, 4, 0, 2, 3, 0, 4, 3], [2, 0, 4, 2, 1, 3, 1, 0, 0], [4, 1, 0, 4, 1, 1, 0, 1, 4], [1, 0, 0, 0, 1, 0, 2, 0, 4]] >;

 
/* Booleans */
RF := recformat< Agroup, Zgroup, abelian, almost_simple, cyclic, metabelian, metacyclic, monomial, nilpotent, perfect, quasisimple, rational, solvable, supersolvable  : BoolElt >; 
booleans_500_27 := rec< RF |  
Agroup := false,
Zgroup := false,
abelian := false,
almost_simple := false,
cyclic := false,
metabelian := false,
metacyclic := false,
monomial := true,
nilpotent := false,
perfect := false,
quasisimple := false,
rational := false,
solvable := true,
supersolvable := true>; 

 
/* Character Table */
G:= GLFp;
C := SequenceToConjugacyClasses([car<Integers(), Integers(), G> |< 1, 1, Matrix(3, [1, 0, 0, 0, 1, 0, 0, 0, 1])>,< 2, 25, Matrix(3, [1, 3, 4, 0, 0, 2, 0, 3, 0])>,< 2, 25, Matrix(3, [1, 0, 0, 4, 1, 1, 2, 0, 4])>,< 2, 25, Matrix(3, [4, 4, 4, 0, 2, 3, 0, 4, 3])>,< 5, 2, Matrix(3, [3, 0, 3, 4, 1, 1, 2, 0, 4])>,< 5, 2, Matrix(3, [0, 0, 1, 3, 1, 2, 4, 0, 2])>,< 5, 10, Matrix(3, [1, 4, 2, 0, 4, 4, 0, 4, 3])>,< 5, 10, Matrix(3, [1, 3, 4, 0, 2, 3, 0, 3, 0])>,< 5, 10, Matrix(3, [4, 0, 2, 2, 1, 3, 3, 0, 3])>,< 5, 10, Matrix(3, [2, 0, 4, 4, 1, 1, 1, 0, 0])>,< 5, 20, Matrix(3, [0, 1, 4, 0, 3, 1, 4, 1, 0])>,< 5, 20, Matrix(3, [0, 2, 2, 2, 0, 0, 4, 2, 3])>,< 5, 20, Matrix(3, [0, 2, 2, 0, 0, 2, 4, 2, 3])>,< 5, 20, Matrix(3, [0, 1, 4, 2, 3, 4, 4, 1, 0])>,< 10, 50, Matrix(3, [2, 3, 3, 2, 0, 0, 1, 3, 4])>,< 10, 50, Matrix(3, [4, 3, 1, 1, 0, 1, 3, 3, 2])>,< 10, 50, Matrix(3, [1, 2, 1, 4, 0, 3, 2, 2, 0])>,< 10, 50, Matrix(3, [1, 1, 3, 4, 3, 2, 2, 1, 2])>,< 10, 50, Matrix(3, [0, 4, 3, 4, 2, 4, 1, 4, 2])>,< 10, 50, Matrix(3, [2, 4, 1, 2, 2, 1, 3, 4, 0])>]); 
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; 
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(5: Sparse := true);
S := [ K |2,0,0,2,2,2,2,2,K.1^2+K.1^-2,K.1+K.1^-1,K.1^2+K.1^-2,K.1+K.1^-1,K.1^2+K.1^-2,K.1+K.1^-1,0,0,0,K.1+K.1^-1,K.1^2+K.1^-2,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |2,0,0,2,2,2,2,2,K.1+K.1^-1,K.1^2+K.1^-2,K.1+K.1^-1,K.1^2+K.1^-2,K.1+K.1^-1,K.1^2+K.1^-2,0,0,0,K.1^2+K.1^-2,K.1+K.1^-1,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |2,0,2,0,2,2,K.1^2+K.1^-2,K.1+K.1^-1,2,2,K.1+K.1^-1,K.1^2+K.1^-2,K.1^2+K.1^-2,K.1+K.1^-1,0,K.1+K.1^-1,K.1^2+K.1^-2,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |2,0,2,0,2,2,K.1+K.1^-1,K.1^2+K.1^-2,2,2,K.1^2+K.1^-2,K.1+K.1^-1,K.1+K.1^-1,K.1^2+K.1^-2,0,K.1^2+K.1^-2,K.1+K.1^-1,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |2,0,-2,0,2,2,K.1^2+K.1^-2,K.1+K.1^-1,2,2,K.1+K.1^-1,K.1^2+K.1^-2,K.1^2+K.1^-2,K.1+K.1^-1,0,-1*K.1-K.1^-1,-1*K.1^2-K.1^-2,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |2,0,-2,0,2,2,K.1+K.1^-1,K.1^2+K.1^-2,2,2,K.1^2+K.1^-2,K.1+K.1^-1,K.1+K.1^-1,K.1^2+K.1^-2,0,-1*K.1^2-K.1^-2,-1*K.1-K.1^-1,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |2,0,0,-2,2,2,2,2,K.1^2+K.1^-2,K.1+K.1^-1,K.1^2+K.1^-2,K.1+K.1^-1,K.1^2+K.1^-2,K.1+K.1^-1,0,0,0,-1*K.1-K.1^-1,-1*K.1^2-K.1^-2,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |2,0,0,-2,2,2,2,2,K.1+K.1^-1,K.1^2+K.1^-2,K.1+K.1^-1,K.1^2+K.1^-2,K.1+K.1^-1,K.1^2+K.1^-2,0,0,0,-1*K.1^2-K.1^-2,-1*K.1-K.1^-1,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |4,0,0,0,4,4,2*K.1^2+2*K.1^-2,2*K.1+2*K.1^-1,2*K.1^2+2*K.1^-2,2*K.1+2*K.1^-1,-1,-1,1-K.1^2-K.1^-2,2+K.1^2+K.1^-2,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |4,0,0,0,4,4,2*K.1+2*K.1^-1,2*K.1^2+2*K.1^-2,2*K.1+2*K.1^-1,2*K.1^2+2*K.1^-2,-1,-1,2+K.1^2+K.1^-2,1-K.1^2-K.1^-2,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |4,0,0,0,4,4,2*K.1^2+2*K.1^-2,2*K.1+2*K.1^-1,2*K.1+2*K.1^-1,2*K.1^2+2*K.1^-2,2+K.1^2+K.1^-2,1-K.1^2-K.1^-2,-1,-1,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |4,0,0,0,4,4,2*K.1+2*K.1^-1,2*K.1^2+2*K.1^-2,2*K.1^2+2*K.1^-2,2*K.1+2*K.1^-1,1-K.1^2-K.1^-2,2+K.1^2+K.1^-2,-1,-1,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |10,2,0,0,5*K.1^2+5*K.1^-2,5*K.1+5*K.1^-1,0,0,0,0,0,0,0,0,K.1+K.1^-1,0,0,0,0,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 |10,2,0,0,5*K.1+5*K.1^-1,5*K.1^2+5*K.1^-2,0,0,0,0,0,0,0,0,K.1^2+K.1^-2,0,0,0,0,K.1+K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(5: Sparse := true);
S := [ K |10,-2,0,0,5*K.1^2+5*K.1^-2,5*K.1+5*K.1^-1,0,0,0,0,0,0,0,0,-1*K.1-K.1^-1,0,0,0,0,-1*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 |10,-2,0,0,5*K.1+5*K.1^-1,5*K.1^2+5*K.1^-2,0,0,0,0,0,0,0,0,-1*K.1^2-K.1^-2,0,0,0,0,-1*K.1-K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
_ := CharacterTable(G : Check := 0); 
chartbl_500_27:= KnownIrreducibles(CR);