/* Group 80.31 downloaded from the LMFDB on 14 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([5, -2, -2, -2, -2, -5, 10, 182, 277, 42, 483, 728, 58, 1204, 809]); a,b := Explode([GPC.1, GPC.3]); AssignNames(~GPC, ["a", "a2", "b", "b2", "b4"]);
GPerm := PermutationGroup< 9 | (2,3,4,5)(6,7)(8,9), (6,8,9,7), (2,4)(3,5), (6,9)(7,8), (1,2,3,5,4) >;
GLZ := MatrixGroup< 6, Integers() | [[-1, -1, -1, -1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1], [0, 0, 1, 0, 0, 0, -1, -1, -1, -1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0]] >;
GLFp := MatrixGroup< 4, GF(3) | [[0, 2, 0, 1, 1, 2, 0, 2, 1, 0, 2, 2, 2, 2, 0, 2], [2, 2, 2, 2, 1, 1, 0, 1, 0, 2, 2, 0, 1, 2, 0, 2], [2, 2, 2, 2, 1, 2, 1, 2, 0, 1, 1, 0, 0, 2, 2, 1], [2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2], [0, 0, 1, 0, 0, 2, 2, 2, 2, 0, 0, 0, 1, 2, 1, 1]] >;

 
/* Booleans */
RF := recformat< Agroup, Zgroup, abelian, almost_simple, cyclic, metabelian, metacyclic, monomial, nilpotent, perfect, quasisimple, rational, solvable, supersolvable  : BoolElt >; 
booleans_80_31 := rec< RF |  
Agroup := false,
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*b^2>,< 2, 5, a^2*b^8>,< 4, 2, b^5>,< 4, 10, a^2*b^3>,< 4, 10, a*b^2>,< 4, 10, a^3*b^6>,< 4, 10, a^3*b^7>,< 4, 10, a*b^19>,< 5, 4, b^4>,< 10, 4, b^2>,< 20, 4, b>,< 20, 4, b^13>]); 
CR := CharacterRing(G);
x := CR!\[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]; 
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]; 
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]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |1,1,-1,-1,-1,-1*K.1,K.1,1,-1*K.1,K.1,1,1,-1,-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,-1*K.1,1,K.1,-1*K.1,1,1,-1,-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,-1*K.1,K.1,-1,K.1,-1*K.1,1,1,1,1]; 
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,-1*K.1,-1,-1*K.1,K.1,1,1,1,1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
x := CR!\[2, -2, -2, 2, 0, 0, 0, 0, 0, 0, 2, -2, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, -2, 2, -2, 0, 0, 0, 0, 0, 0, 2, -2, 0, 0]; 
x`IsCharacter := true;
x`Schur := -1;
x`IsIrreducible := true; 
x := CR!\[4, 4, 0, 0, 4, 0, 0, 0, 0, 0, -1, -1, -1, -1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[4, 4, 0, 0, -4, 0, 0, 0, 0, 0, -1, -1, 1, 1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(20: Sparse := true);
S := [ K |4,-4,0,0,0,0,0,0,0,0,-1,1,-2*K.1^3+K.1^5-2*K.1^7,2*K.1^3-K.1^5+2*K.1^7]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(20: Sparse := true);
S := [ K |4,-4,0,0,0,0,0,0,0,0,-1,1,2*K.1^3-K.1^5+2*K.1^7,-2*K.1^3+K.1^5-2*K.1^7]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
_ := CharacterTable(G : Check := 0); 
chartbl_80_31:= KnownIrreducibles(CR);