/* Group 112.16 downloaded from the LMFDB on 03 October 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([5, -2, -2, -2, -2, -7, 40, 61, 26, 182, 42, 1209]); a,b,c := Explode([GPC.1, GPC.2, GPC.5]); AssignNames(~GPC, ["a", "b", "b2", "b4", "c"]);
GPerm := PermutationGroup< 15 | (2,5)(3,7)(6,8)(10,11)(12,13)(14,15), (1,2,4,6)(3,8,7,5), (1,3,4,7)(2,5,6,8), (1,4)(2,6)(3,7)(5,8), (9,10,12,14,15,13,11) >;
GLZN := MatrixGroup< 2, Integers(21) | [[1, 3, 0, 1], [20, 14, 14, 13], [8, 18, 7, 13], [1, 14, 14, 8], [8, 0, 0, 8]] >;

 
/* Booleans */
RF := recformat< Agroup, Zgroup, abelian, almost_simple, cyclic, metabelian, metacyclic, monomial, nilpotent, perfect, quasisimple, rational, solvable, supersolvable  : BoolElt >; 
booleans_112_16 := rec< RF |  
Agroup := false,
Zgroup := false,
abelian := false,
almost_simple := false,
cyclic := false,
metabelian := true,
metacyclic := false,
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^4>,< 2, 28, a*b^5*c^2>,< 4, 2, b^6>,< 4, 4, a*b^4>,< 7, 2, c^4>,< 7, 2, c>,< 7, 2, c^5>,< 8, 14, b^7*c^2>,< 8, 14, b*c^2>,< 14, 2, b^4*c^2>,< 14, 2, b^4*c^6>,< 14, 2, b^4*c^3>,< 28, 4, b^2*c>,< 28, 4, b^2*c^3>,< 28, 4, b^2*c^2>,< 28, 4, a*c>,< 28, 4, a*c^6>,< 28, 4, a*c^3>,< 28, 4, a*c^4>,< 28, 4, a*c^5>,< 28, 4, a*c^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]; 
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, 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, 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, 1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 2, 0, -2, 0, 2, 2, 2, 0, 0, 2, 2, 2, 0, -2, 0, -2, 0, 0, 0, 0, -2]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(8: Sparse := true);
S := [ K |2,-2,0,0,0,2,2,2,-1*K.1-K.1^3,K.1+K.1^3,-2,-2,-2,0,0,0,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(8: Sparse := true);
S := [ K |2,-2,0,0,0,2,2,2,K.1+K.1^3,-1*K.1-K.1^3,-2,-2,-2,0,0,0,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(7: Sparse := true);
S := [ K |2,2,0,2,2,K.1^3+K.1^-3,K.1+K.1^-1,K.1^2+K.1^-2,0,0,K.1^2+K.1^-2,K.1+K.1^-1,K.1^3+K.1^-3,K.1+K.1^-1,K.1^3+K.1^-3,K.1^2+K.1^-2,K.1+K.1^-1,K.1^3+K.1^-3,K.1^2+K.1^-2,K.1+K.1^-1,K.1^3+K.1^-3,K.1^2+K.1^-2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(7: Sparse := true);
S := [ K |2,2,0,2,2,K.1^2+K.1^-2,K.1^3+K.1^-3,K.1+K.1^-1,0,0,K.1+K.1^-1,K.1^3+K.1^-3,K.1^2+K.1^-2,K.1^3+K.1^-3,K.1^2+K.1^-2,K.1+K.1^-1,K.1^3+K.1^-3,K.1^2+K.1^-2,K.1+K.1^-1,K.1^3+K.1^-3,K.1^2+K.1^-2,K.1+K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(7: Sparse := true);
S := [ K |2,2,0,2,2,K.1+K.1^-1,K.1^2+K.1^-2,K.1^3+K.1^-3,0,0,K.1^3+K.1^-3,K.1^2+K.1^-2,K.1+K.1^-1,K.1^2+K.1^-2,K.1+K.1^-1,K.1^3+K.1^-3,K.1^2+K.1^-2,K.1+K.1^-1,K.1^3+K.1^-3,K.1^2+K.1^-2,K.1+K.1^-1,K.1^3+K.1^-3]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(7: Sparse := true);
S := [ K |2,2,0,2,-2,K.1^3+K.1^-3,K.1+K.1^-1,K.1^2+K.1^-2,0,0,K.1^2+K.1^-2,K.1+K.1^-1,K.1^3+K.1^-3,-1*K.1-K.1^-1,K.1^3+K.1^-3,-1*K.1^2-K.1^-2,K.1+K.1^-1,-1*K.1^3-K.1^-3,-1*K.1^2-K.1^-2,-1*K.1-K.1^-1,-1*K.1^3-K.1^-3,K.1^2+K.1^-2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(7: Sparse := true);
S := [ K |2,2,0,2,-2,K.1^2+K.1^-2,K.1^3+K.1^-3,K.1+K.1^-1,0,0,K.1+K.1^-1,K.1^3+K.1^-3,K.1^2+K.1^-2,-1*K.1^3-K.1^-3,K.1^2+K.1^-2,-1*K.1-K.1^-1,K.1^3+K.1^-3,-1*K.1^2-K.1^-2,-1*K.1-K.1^-1,-1*K.1^3-K.1^-3,-1*K.1^2-K.1^-2,K.1+K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(7: Sparse := true);
S := [ K |2,2,0,2,-2,K.1+K.1^-1,K.1^2+K.1^-2,K.1^3+K.1^-3,0,0,K.1^3+K.1^-3,K.1^2+K.1^-2,K.1+K.1^-1,-1*K.1^2-K.1^-2,K.1+K.1^-1,-1*K.1^3-K.1^-3,K.1^2+K.1^-2,-1*K.1-K.1^-1,-1*K.1^3-K.1^-3,-1*K.1^2-K.1^-2,-1*K.1-K.1^-1,K.1^3+K.1^-3]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(7: Sparse := true);
S := [ K |2,2,0,-2,0,K.1^3+K.1^-3,K.1+K.1^-1,K.1^2+K.1^-2,0,0,K.1^2+K.1^-2,K.1+K.1^-1,K.1^3+K.1^-3,-1*K.1+K.1^-1,-1*K.1^3-K.1^-3,-1*K.1^2+K.1^-2,-1*K.1-K.1^-1,K.1^3-K.1^-3,K.1^2-K.1^-2,K.1-K.1^-1,-1*K.1^3+K.1^-3,-1*K.1^2-K.1^-2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(7: Sparse := true);
S := [ K |2,2,0,-2,0,K.1^3+K.1^-3,K.1+K.1^-1,K.1^2+K.1^-2,0,0,K.1^2+K.1^-2,K.1+K.1^-1,K.1^3+K.1^-3,K.1-K.1^-1,-1*K.1^3-K.1^-3,K.1^2-K.1^-2,-1*K.1-K.1^-1,-1*K.1^3+K.1^-3,-1*K.1^2+K.1^-2,-1*K.1+K.1^-1,K.1^3-K.1^-3,-1*K.1^2-K.1^-2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(7: Sparse := true);
S := [ K |2,2,0,-2,0,K.1^2+K.1^-2,K.1^3+K.1^-3,K.1+K.1^-1,0,0,K.1+K.1^-1,K.1^3+K.1^-3,K.1^2+K.1^-2,-1*K.1^3+K.1^-3,-1*K.1^2-K.1^-2,K.1-K.1^-1,-1*K.1^3-K.1^-3,K.1^2-K.1^-2,-1*K.1+K.1^-1,K.1^3-K.1^-3,-1*K.1^2+K.1^-2,-1*K.1-K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(7: Sparse := true);
S := [ K |2,2,0,-2,0,K.1^2+K.1^-2,K.1^3+K.1^-3,K.1+K.1^-1,0,0,K.1+K.1^-1,K.1^3+K.1^-3,K.1^2+K.1^-2,K.1^3-K.1^-3,-1*K.1^2-K.1^-2,-1*K.1+K.1^-1,-1*K.1^3-K.1^-3,-1*K.1^2+K.1^-2,K.1-K.1^-1,-1*K.1^3+K.1^-3,K.1^2-K.1^-2,-1*K.1-K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(7: Sparse := true);
S := [ K |2,2,0,-2,0,K.1+K.1^-1,K.1^2+K.1^-2,K.1^3+K.1^-3,0,0,K.1^3+K.1^-3,K.1^2+K.1^-2,K.1+K.1^-1,-1*K.1^2+K.1^-2,-1*K.1-K.1^-1,K.1^3-K.1^-3,-1*K.1^2-K.1^-2,-1*K.1+K.1^-1,-1*K.1^3+K.1^-3,K.1^2-K.1^-2,K.1-K.1^-1,-1*K.1^3-K.1^-3]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(7: Sparse := true);
S := [ K |2,2,0,-2,0,K.1+K.1^-1,K.1^2+K.1^-2,K.1^3+K.1^-3,0,0,K.1^3+K.1^-3,K.1^2+K.1^-2,K.1+K.1^-1,K.1^2-K.1^-2,-1*K.1-K.1^-1,-1*K.1^3+K.1^-3,-1*K.1^2-K.1^-2,K.1-K.1^-1,K.1^3-K.1^-3,-1*K.1^2+K.1^-2,-1*K.1+K.1^-1,-1*K.1^3-K.1^-3]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(7: Sparse := true);
S := [ K |4,-4,0,0,0,2*K.1^3+2*K.1^-3,2*K.1+2*K.1^-1,2*K.1^2+2*K.1^-2,0,0,-2*K.1^2-2*K.1^-2,-2*K.1-2*K.1^-1,-2*K.1^3-2*K.1^-3,0,0,0,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(7: Sparse := true);
S := [ K |4,-4,0,0,0,2*K.1^2+2*K.1^-2,2*K.1^3+2*K.1^-3,2*K.1+2*K.1^-1,0,0,-2*K.1-2*K.1^-1,-2*K.1^3-2*K.1^-3,-2*K.1^2-2*K.1^-2,0,0,0,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(7: Sparse := true);
S := [ K |4,-4,0,0,0,2*K.1+2*K.1^-1,2*K.1^2+2*K.1^-2,2*K.1^3+2*K.1^-3,0,0,-2*K.1^3-2*K.1^-3,-2*K.1^2-2*K.1^-2,-2*K.1-2*K.1^-1,0,0,0,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
_ := CharacterTable(G : Check := 0); 
chartbl_112_16:= KnownIrreducibles(CR);