/* Group 384.1726 downloaded from the LMFDB on 24 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([8, 2, 2, 2, 2, 2, 2, 2, 3, 41, 3467, 211, 91, 8453, 4045, 2709, 141, 17926, 9422, 166, 16391, 8207]); a,b,c,d := Explode([GPC.1, GPC.2, GPC.4, GPC.6]); AssignNames(~GPC, ["a", "b", "b2", "c", "c2", "d", "d2", "d4"]);
GPerm := PermutationGroup< 19 | (1,2,4,6)(3,7,11,13)(5,9,12,8)(10,15,16,14), (2,6)(3,10)(4,12)(7,14)(8,9)(13,15), (1,3)(2,7)(4,11)(5,10)(6,13)(8,14)(9,15)(12,16)(18,19), (1,4,5,12)(2,8,9,6)(3,11,10,16)(7,14,15,13), (1,4)(2,6)(3,11)(5,12)(7,13)(8,9)(10,16)(14,15), (2,9)(6,8)(7,15)(13,14), (1,5)(2,9)(3,10)(4,12)(6,8)(7,15)(11,16)(13,14), (17,18,19) >;

 
/* Booleans */
RF := recformat< Agroup, Zgroup, abelian, almost_simple, cyclic, metabelian, metacyclic, monomial, nilpotent, perfect, quasisimple, rational, solvable, supersolvable  : BoolElt >; 
booleans_384_1726 := 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, c^2>,< 2, 2, d^6>,< 2, 4, b^2>,< 2, 4, b^2*c^3*d^6>,< 2, 6, a>,< 2, 6, a*d^2>,< 2, 8, b^2*d^9>,< 2, 12, a*b^2>,< 2, 12, a*b^2*c>,< 3, 2, d^8>,< 4, 4, c^3>,< 4, 8, d^3>,< 4, 8, a*b^3>,< 4, 8, a*b>,< 4, 8, a*b^3*c^3*d^9>,< 4, 8, a*b*d^9>,< 4, 12, a*c>,< 4, 24, a*c*d^9>,< 4, 24, a*b^2*d^9>,< 4, 24, b*c^2*d^3>,< 4, 24, b^3*c*d^3>,< 4, 24, b^3*c*d^8>,< 4, 24, b*c*d^2>,< 6, 2, c^2*d^8>,< 6, 4, d^2>,< 6, 8, b^2*d^8>,< 6, 8, b^2*c^3*d^2>,< 6, 16, b^2*d>,< 12, 8, c*d^4>,< 12, 8, d>,< 12, 8, d^5>,< 12, 16, a*b*d^4>,< 12, 16, a*b^3*d^4>,< 12, 16, a*b*d>,< 12, 16, a*b^3*d>]); 
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, 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, 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, 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, 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, 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, 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, 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, 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,-1,-1,1,1,1,1,-1*K.1,-1*K.1,K.1,1,K.1,-1,-1*K.1,1,-1*K.1,-1,K.1,K.1,1,1,-1,-1,-1,1,1,1,K.1,-1*K.1,K.1,-1*K.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,-1,-1,1,1,1,1,K.1,K.1,-1*K.1,1,-1*K.1,-1,K.1,1,K.1,-1,-1*K.1,-1*K.1,1,1,-1,-1,-1,1,1,1,-1*K.1,K.1,-1*K.1,K.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,-1,1,1,1,1,1,-1*K.1,K.1,K.1,-1,-1*K.1,-1,K.1,-1,-1*K.1,1,-1*K.1,K.1,1,1,-1,-1,1,-1,1,-1,K.1,-1*K.1,-1*K.1,K.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,-1,1,1,1,1,1,K.1,-1*K.1,-1*K.1,-1,K.1,-1,-1*K.1,-1,K.1,1,K.1,-1*K.1,1,1,-1,-1,1,-1,1,-1,-1*K.1,K.1,K.1,-1*K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |1,1,1,-1,-1,1,1,-1,-1,-1,1,1,-1*K.1,-1*K.1,K.1,1,K.1,1,K.1,-1,K.1,1,-1*K.1,-1*K.1,1,1,-1,-1,-1,1,1,1,K.1,-1*K.1,K.1,-1*K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |1,1,1,-1,-1,1,1,-1,-1,-1,1,1,K.1,K.1,-1*K.1,1,-1*K.1,1,-1*K.1,-1,-1*K.1,1,K.1,K.1,1,1,-1,-1,-1,1,1,1,-1*K.1,K.1,-1*K.1,K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |1,1,1,-1,-1,1,1,1,-1,-1,1,1,-1*K.1,K.1,K.1,-1,-1*K.1,1,-1*K.1,1,K.1,-1,K.1,-1*K.1,1,1,-1,-1,1,-1,1,-1,K.1,-1*K.1,-1*K.1,K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |1,1,1,-1,-1,1,1,1,-1,-1,1,1,K.1,-1*K.1,-1*K.1,-1,K.1,1,K.1,1,-1*K.1,-1,-1*K.1,K.1,1,1,-1,-1,1,-1,1,-1,-1*K.1,K.1,K.1,-1*K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
x := CR!\[2, 2, 2, 2, 2, 0, 0, 2, 0, 0, -1, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, -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, -2, 2, -2, -2, 0, 2, -2, 2, -2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 2, -2, 2, 0, 0, -2, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 2, 2, -2, 2, 2, 2, 0, -2, 2, 2, -2, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 2, 2, -2, 2, 0, 0, -2, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 2, 2, 2, -2, -2, -2, 0, -2, 2, 2, -2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 2, 2, -2, 0, 0, -2, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 2, 2, 2, -2, 2, 2, 0, 2, -2, 2, -2, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 2, 2, 2, -2, 0, 0, -2, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 2, 2, 2, 2, 0, 0, -2, 0, 0, -1, 2, -2, 2, -2, -2, 2, 0, 0, 0, 0, 0, 0, 0, -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, 2, 2, 0, 0, -2, 0, 0, -1, 2, 2, -2, 2, -2, -2, 0, 0, 0, 0, 0, 0, 0, -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, 2, 2, 0, 0, 2, 0, 0, -1, 2, -2, -2, -2, 2, -2, 0, 0, 0, 0, 0, 0, 0, -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 |2,2,2,-2,-2,0,0,-2,0,0,-1,2,-2*K.1,-2*K.1,2*K.1,2,2*K.1,0,0,0,0,0,0,0,-1,-1,1,1,1,-1,-1,-1,-1*K.1,K.1,-1*K.1,K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |2,2,2,-2,-2,0,0,-2,0,0,-1,2,2*K.1,2*K.1,-2*K.1,2,-2*K.1,0,0,0,0,0,0,0,-1,-1,1,1,1,-1,-1,-1,K.1,-1*K.1,K.1,-1*K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |2,2,2,-2,-2,0,0,2,0,0,-1,2,-2*K.1,2*K.1,2*K.1,-2,-2*K.1,0,0,0,0,0,0,0,-1,-1,1,1,-1,1,-1,1,-1*K.1,K.1,K.1,-1*K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |2,2,2,-2,-2,0,0,2,0,0,-1,2,2*K.1,-2*K.1,-2*K.1,-2,2*K.1,0,0,0,0,0,0,0,-1,-1,1,1,-1,1,-1,1,K.1,-1*K.1,-1*K.1,K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
x := CR!\[4, 4, -4, 0, 0, -4, 4, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[4, 4, -4, 0, 0, 4, -4, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[4, 4, 4, -4, 4, 0, 0, 0, 0, 0, -2, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 2, -2, 0, 0, 2, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[4, 4, 4, 4, -4, 0, 0, 0, 0, 0, -2, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -2, -2, 2, 0, 0, 2, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[8, -8, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[8, 8, -8, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |8,-8,0,0,0,0,0,0,0,0,-4,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,-2-4*K.1,0,2+4*K.1,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |8,-8,0,0,0,0,0,0,0,0,-4,0,0,0,0,0,0,0,0,0,0,0,0,0,4,0,0,0,0,2+4*K.1,0,-2-4*K.1,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
_ := CharacterTable(G : Check := 0); 
chartbl_384_1726:= KnownIrreducibles(CR);