/* Group 168.16 downloaded from the LMFDB on 06 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, -7, -3, 541, 26, 782, 42, 963, 1409]); a,b,c := Explode([GPC.1, GPC.2, GPC.5]); AssignNames(~GPC, ["a", "b", "b2", "b4", "c"]);
GPerm := PermutationGroup< 14 | (1,2)(3,4)(9,10)(11,12)(13,14), (1,3,4,2)(6,7), (1,4)(2,3), (5,6,7), (8,9,11,13,14,12,10) >;
GLZN := MatrixGroup< 2, Integers(28) | [[22, 23, 7, 6], [15, 0, 0, 15], [1, 4, 0, 1], [22, 7, 7, 1], [13, 14, 7, 27]] >;

 
/* Booleans */
RF := recformat< Agroup, Zgroup, abelian, almost_simple, cyclic, metabelian, metacyclic, monomial, nilpotent, perfect, quasisimple, rational, solvable, supersolvable  : BoolElt >; 
booleans_168_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^14>,< 2, 14, a*b^20>,< 2, 42, a*b^25*c>,< 3, 2, c^2>,< 4, 6, b^7>,< 6, 2, b^14*c>,< 6, 14, a*b^20*c^2>,< 6, 14, a*b^20*c>,< 7, 2, b^4>,< 7, 2, b^8>,< 7, 2, b^12>,< 14, 2, b^2>,< 14, 2, b^6>,< 14, 2, b^10>,< 21, 4, b^4*c^2>,< 21, 4, b^8*c>,< 21, 4, b^16*c^2>,< 28, 6, b>,< 28, 6, b^3>,< 28, 6, b^5>,< 28, 6, b^9>,< 28, 6, b^17>,< 28, 6, b^13>,< 42, 4, b^2*c>,< 42, 4, b^10*c>,< 42, 4, b^6*c>]); 
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]; 
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, 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, -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, -1, -1, 1, 1, 1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 2, 2, 0, -1, 0, -1, -1, -1, 2, 2, 2, 2, 2, 2, -1, -1, -1, 0, 0, 0, 0, 0, 0, -1, -1, -1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, -2, 0, 0, 2, 0, -2, 0, 0, 2, 2, 2, -2, -2, -2, 2, 2, 2, 0, 0, 0, 0, 0, 0, -2, -2, -2]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 2, -2, 0, -1, 0, -1, 1, 1, 2, 2, 2, 2, 2, 2, -1, -1, -1, 0, 0, 0, 0, 0, 0, -1, -1, -1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |2,-2,0,0,-1,0,1,-1-2*K.1,1+2*K.1,2,2,2,-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,1,1,1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |2,-2,0,0,-1,0,1,1+2*K.1,-1-2*K.1,2,2,2,-2,-2,-2,-1,-1,-1,0,0,0,0,0,0,1,1,1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(7: Sparse := true);
S := [ K |2,2,0,0,2,2,2,0,0,K.1^3+K.1^-3,K.1+K.1^-1,K.1^2+K.1^-2,K.1^2+K.1^-2,K.1+K.1^-1,K.1^3+K.1^-3,K.1+K.1^-1,K.1^2+K.1^-2,K.1^3+K.1^-3,K.1+K.1^-1,K.1^3+K.1^-3,K.1^2+K.1^-2,K.1^2+K.1^-2,K.1^3+K.1^-3,K.1+K.1^-1,K.1^2+K.1^-2,K.1^3+K.1^-3,K.1+K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(7: Sparse := true);
S := [ K |2,2,0,0,2,2,2,0,0,K.1^2+K.1^-2,K.1^3+K.1^-3,K.1+K.1^-1,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+K.1^-1,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+K.1^-1,K.1^2+K.1^-2,K.1^3+K.1^-3,K.1+K.1^-1,K.1^2+K.1^-2,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 |2,2,0,0,2,2,2,0,0,K.1+K.1^-1,K.1^2+K.1^-2,K.1^3+K.1^-3,K.1^3+K.1^-3,K.1^2+K.1^-2,K.1+K.1^-1,K.1^2+K.1^-2,K.1^3+K.1^-3,K.1+K.1^-1,K.1^2+K.1^-2,K.1+K.1^-1,K.1^3+K.1^-3,K.1^3+K.1^-3,K.1+K.1^-1,K.1^2+K.1^-2,K.1^3+K.1^-3,K.1+K.1^-1,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 |2,2,0,0,2,-2,2,0,0,K.1^3+K.1^-3,K.1+K.1^-1,K.1^2+K.1^-2,K.1^2+K.1^-2,K.1+K.1^-1,K.1^3+K.1^-3,K.1+K.1^-1,K.1^2+K.1^-2,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.1^2-K.1^-2,-1*K.1^3-K.1^-3,-1*K.1-K.1^-1,K.1^2+K.1^-2,K.1^3+K.1^-3,K.1+K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(7: Sparse := true);
S := [ K |2,2,0,0,2,-2,2,0,0,K.1^2+K.1^-2,K.1^3+K.1^-3,K.1+K.1^-1,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+K.1^-1,K.1^2+K.1^-2,-1*K.1^3-K.1^-3,-1*K.1^2-K.1^-2,-1*K.1-K.1^-1,-1*K.1-K.1^-1,-1*K.1^2-K.1^-2,-1*K.1^3-K.1^-3,K.1+K.1^-1,K.1^2+K.1^-2,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 |2,2,0,0,2,-2,2,0,0,K.1+K.1^-1,K.1^2+K.1^-2,K.1^3+K.1^-3,K.1^3+K.1^-3,K.1^2+K.1^-2,K.1+K.1^-1,K.1^2+K.1^-2,K.1^3+K.1^-3,K.1+K.1^-1,-1*K.1^2-K.1^-2,-1*K.1-K.1^-1,-1*K.1^3-K.1^-3,-1*K.1^3-K.1^-3,-1*K.1-K.1^-1,-1*K.1^2-K.1^-2,K.1^3+K.1^-3,K.1+K.1^-1,K.1^2+K.1^-2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(28: Sparse := true);
S := [ K |2,-2,0,0,2,0,-2,0,0,-1*K.1^2-K.1^-2,K.1^4+K.1^-4,-1*K.1^6-K.1^-6,K.1^6+K.1^-6,-1*K.1^4-K.1^-4,K.1^2+K.1^-2,K.1^4+K.1^-4,-1*K.1^6-K.1^-6,-1*K.1^2-K.1^-2,-1*K.1^3-K.1^-3,-1*K.1^5-K.1^-5,-1*K.1-K.1^-1,K.1+K.1^-1,K.1^5+K.1^-5,K.1^3+K.1^-3,K.1^6+K.1^-6,K.1^2+K.1^-2,-1*K.1^4-K.1^-4]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(28: Sparse := true);
S := [ K |2,-2,0,0,2,0,-2,0,0,-1*K.1^2-K.1^-2,K.1^4+K.1^-4,-1*K.1^6-K.1^-6,K.1^6+K.1^-6,-1*K.1^4-K.1^-4,K.1^2+K.1^-2,K.1^4+K.1^-4,-1*K.1^6-K.1^-6,-1*K.1^2-K.1^-2,K.1^3+K.1^-3,K.1^5+K.1^-5,K.1+K.1^-1,-1*K.1-K.1^-1,-1*K.1^5-K.1^-5,-1*K.1^3-K.1^-3,K.1^6+K.1^-6,K.1^2+K.1^-2,-1*K.1^4-K.1^-4]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(28: Sparse := true);
S := [ K |2,-2,0,0,2,0,-2,0,0,-1*K.1^6-K.1^-6,-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.1^6+K.1^-6,-1*K.1^2-K.1^-2,K.1^4+K.1^-4,-1*K.1^6-K.1^-6,-1*K.1^5-K.1^-5,-1*K.1-K.1^-1,K.1^3+K.1^-3,-1*K.1^3-K.1^-3,K.1+K.1^-1,K.1^5+K.1^-5,-1*K.1^4-K.1^-4,K.1^6+K.1^-6,K.1^2+K.1^-2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(28: Sparse := true);
S := [ K |2,-2,0,0,2,0,-2,0,0,-1*K.1^6-K.1^-6,-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.1^6+K.1^-6,-1*K.1^2-K.1^-2,K.1^4+K.1^-4,-1*K.1^6-K.1^-6,K.1^5+K.1^-5,K.1+K.1^-1,-1*K.1^3-K.1^-3,K.1^3+K.1^-3,-1*K.1-K.1^-1,-1*K.1^5-K.1^-5,-1*K.1^4-K.1^-4,K.1^6+K.1^-6,K.1^2+K.1^-2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(28: Sparse := true);
S := [ K |2,-2,0,0,2,0,-2,0,0,K.1^4+K.1^-4,-1*K.1^6-K.1^-6,-1*K.1^2-K.1^-2,K.1^2+K.1^-2,K.1^6+K.1^-6,-1*K.1^4-K.1^-4,-1*K.1^6-K.1^-6,-1*K.1^2-K.1^-2,K.1^4+K.1^-4,-1*K.1-K.1^-1,K.1^3+K.1^-3,K.1^5+K.1^-5,-1*K.1^5-K.1^-5,-1*K.1^3-K.1^-3,K.1+K.1^-1,K.1^2+K.1^-2,-1*K.1^4-K.1^-4,K.1^6+K.1^-6]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(28: Sparse := true);
S := [ K |2,-2,0,0,2,0,-2,0,0,K.1^4+K.1^-4,-1*K.1^6-K.1^-6,-1*K.1^2-K.1^-2,K.1^2+K.1^-2,K.1^6+K.1^-6,-1*K.1^4-K.1^-4,-1*K.1^6-K.1^-6,-1*K.1^2-K.1^-2,K.1^4+K.1^-4,K.1+K.1^-1,-1*K.1^3-K.1^-3,-1*K.1^5-K.1^-5,K.1^5+K.1^-5,K.1^3+K.1^-3,-1*K.1-K.1^-1,K.1^2+K.1^-2,-1*K.1^4-K.1^-4,K.1^6+K.1^-6]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(7: Sparse := true);
S := [ K |4,4,0,0,-2,0,-2,0,0,2*K.1^3+2*K.1^-3,2*K.1+2*K.1^-1,2*K.1^2+2*K.1^-2,2*K.1^2+2*K.1^-2,2*K.1+2*K.1^-1,2*K.1^3+2*K.1^-3,-1*K.1-K.1^-1,-1*K.1^2-K.1^-2,-1*K.1^3-K.1^-3,0,0,0,0,0,0,-1*K.1^2-K.1^-2,-1*K.1^3-K.1^-3,-1*K.1-K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(7: Sparse := true);
S := [ K |4,4,0,0,-2,0,-2,0,0,2*K.1^2+2*K.1^-2,2*K.1^3+2*K.1^-3,2*K.1+2*K.1^-1,2*K.1+2*K.1^-1,2*K.1^3+2*K.1^-3,2*K.1^2+2*K.1^-2,-1*K.1^3-K.1^-3,-1*K.1-K.1^-1,-1*K.1^2-K.1^-2,0,0,0,0,0,0,-1*K.1-K.1^-1,-1*K.1^2-K.1^-2,-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 |4,4,0,0,-2,0,-2,0,0,2*K.1+2*K.1^-1,2*K.1^2+2*K.1^-2,2*K.1^3+2*K.1^-3,2*K.1^3+2*K.1^-3,2*K.1^2+2*K.1^-2,2*K.1+2*K.1^-1,-1*K.1^2-K.1^-2,-1*K.1^3-K.1^-3,-1*K.1-K.1^-1,0,0,0,0,0,0,-1*K.1^3-K.1^-3,-1*K.1-K.1^-1,-1*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 |4,-4,0,0,-2,0,2,0,0,2*K.1^3+2*K.1^-3,2*K.1+2*K.1^-1,2*K.1^2+2*K.1^-2,-2*K.1^2-2*K.1^-2,-2*K.1-2*K.1^-1,-2*K.1^3-2*K.1^-3,-1*K.1-K.1^-1,-1*K.1^2-K.1^-2,-1*K.1^3-K.1^-3,0,0,0,0,0,0,K.1^2+K.1^-2,K.1^3+K.1^-3,K.1+K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(7: Sparse := true);
S := [ K |4,-4,0,0,-2,0,2,0,0,2*K.1^2+2*K.1^-2,2*K.1^3+2*K.1^-3,2*K.1+2*K.1^-1,-2*K.1-2*K.1^-1,-2*K.1^3-2*K.1^-3,-2*K.1^2-2*K.1^-2,-1*K.1^3-K.1^-3,-1*K.1-K.1^-1,-1*K.1^2-K.1^-2,0,0,0,0,0,0,K.1+K.1^-1,K.1^2+K.1^-2,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 |4,-4,0,0,-2,0,2,0,0,2*K.1+2*K.1^-1,2*K.1^2+2*K.1^-2,2*K.1^3+2*K.1^-3,-2*K.1^3-2*K.1^-3,-2*K.1^2-2*K.1^-2,-2*K.1-2*K.1^-1,-1*K.1^2-K.1^-2,-1*K.1^3-K.1^-3,-1*K.1-K.1^-1,0,0,0,0,0,0,K.1^3+K.1^-3,K.1+K.1^-1,K.1^2+K.1^-2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
_ := CharacterTable(G : Check := 0); 
chartbl_168_16:= KnownIrreducibles(CR);