/* Group 48.1 downloaded from the LMFDB on 25 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, -3, 10, 26, 42, 804]); a,b := Explode([GPC.1, GPC.5]); AssignNames(~GPC, ["a", "a2", "a4", "a8", "b"]);
GPerm := PermutationGroup< 19 | (1,2,3,6,4,7,9,12,5,8,10,13,11,14,15,16)(18,19), (1,3,4,9,5,10,11,15)(2,6,7,12,8,13,14,16), (1,4,5,11)(2,7,8,14)(3,9,10,15)(6,12,13,16), (1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16), (17,18,19) >;
GLFp := MatrixGroup< 2, GF(17) | [[2, 11, 2, 15], [13, 3, 1, 13]] >;

 
/* Booleans */
RF := recformat< Agroup, Zgroup, abelian, almost_simple, cyclic, metabelian, metacyclic, monomial, nilpotent, perfect, quasisimple, rational, solvable, supersolvable  : BoolElt >; 
booleans_48_1 := rec< RF |  
Agroup := true,
Zgroup := true,
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, a^8>,< 3, 2, b^2>,< 4, 1, a^4>,< 4, 1, a^12>,< 6, 2, a^8*b>,< 8, 1, a^2>,< 8, 1, a^14>,< 8, 1, a^6>,< 8, 1, a^10>,< 12, 2, a^4*b^2>,< 12, 2, a^12*b>,< 16, 3, a>,< 16, 3, a^15>,< 16, 3, a^3>,< 16, 3, a^13>,< 16, 3, a^5>,< 16, 3, a^11>,< 16, 3, a^7>,< 16, 3, a^9>,< 24, 2, a^2*b>,< 24, 2, a^14*b>,< 24, 2, a^10*b>,< 24, 2, a^6*b>]); 
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]; 
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]; 
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*K.1,K.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 |1,1,1,1,1,1,-1,-1,-1,-1,1,1,K.1,K.1,-1*K.1,K.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(8: Sparse := true);
S := [ K |1,1,1,-1,-1,1,-1*K.1^2,K.1^2,K.1^2,-1*K.1^2,-1,-1,K.1^3,K.1^3,K.1,-1*K.1^3,-1*K.1^3,K.1,-1*K.1,-1*K.1,-1*K.1^2,K.1^2,-1*K.1^2,K.1^2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(8: Sparse := true);
S := [ K |1,1,1,-1,-1,1,K.1^2,-1*K.1^2,-1*K.1^2,K.1^2,-1,-1,-1*K.1,-1*K.1,-1*K.1^3,K.1,K.1,-1*K.1^3,K.1^3,K.1^3,K.1^2,-1*K.1^2,K.1^2,-1*K.1^2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(8: Sparse := true);
S := [ K |1,1,1,-1,-1,1,-1*K.1^2,K.1^2,K.1^2,-1*K.1^2,-1,-1,-1*K.1^3,-1*K.1^3,-1*K.1,K.1^3,K.1^3,-1*K.1,K.1,K.1,-1*K.1^2,K.1^2,-1*K.1^2,K.1^2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(8: Sparse := true);
S := [ K |1,1,1,-1,-1,1,K.1^2,-1*K.1^2,-1*K.1^2,K.1^2,-1,-1,K.1,K.1,K.1^3,-1*K.1,-1*K.1,K.1^3,-1*K.1^3,-1*K.1^3,K.1^2,-1*K.1^2,K.1^2,-1*K.1^2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(16: Sparse := true);
S := [ K |1,-1,1,-1*K.1^4,K.1^4,-1,K.1^6,-1*K.1^2,K.1^2,-1*K.1^6,-1*K.1^4,K.1^4,-1*K.1^3,K.1^3,-1*K.1,K.1^7,-1*K.1^7,K.1,K.1^5,-1*K.1^5,K.1^6,-1*K.1^2,-1*K.1^6,K.1^2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(16: Sparse := true);
S := [ K |1,-1,1,K.1^4,-1*K.1^4,-1,-1*K.1^2,K.1^6,-1*K.1^6,K.1^2,K.1^4,-1*K.1^4,K.1^5,-1*K.1^5,K.1^7,-1*K.1,K.1,-1*K.1^7,-1*K.1^3,K.1^3,-1*K.1^2,K.1^6,K.1^2,-1*K.1^6]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(16: Sparse := true);
S := [ K |1,-1,1,-1*K.1^4,K.1^4,-1,K.1^6,-1*K.1^2,K.1^2,-1*K.1^6,-1*K.1^4,K.1^4,K.1^3,-1*K.1^3,K.1,-1*K.1^7,K.1^7,-1*K.1,-1*K.1^5,K.1^5,K.1^6,-1*K.1^2,-1*K.1^6,K.1^2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(16: Sparse := true);
S := [ K |1,-1,1,K.1^4,-1*K.1^4,-1,-1*K.1^2,K.1^6,-1*K.1^6,K.1^2,K.1^4,-1*K.1^4,-1*K.1^5,K.1^5,-1*K.1^7,K.1,-1*K.1,K.1^7,K.1^3,-1*K.1^3,-1*K.1^2,K.1^6,K.1^2,-1*K.1^6]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(16: Sparse := true);
S := [ K |1,-1,1,-1*K.1^4,K.1^4,-1,-1*K.1^6,K.1^2,-1*K.1^2,K.1^6,-1*K.1^4,K.1^4,K.1^7,-1*K.1^7,-1*K.1^5,K.1^3,-1*K.1^3,K.1^5,-1*K.1,K.1,-1*K.1^6,K.1^2,K.1^6,-1*K.1^2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(16: Sparse := true);
S := [ K |1,-1,1,K.1^4,-1*K.1^4,-1,K.1^2,-1*K.1^6,K.1^6,-1*K.1^2,K.1^4,-1*K.1^4,-1*K.1,K.1,K.1^3,-1*K.1^5,K.1^5,-1*K.1^3,K.1^7,-1*K.1^7,K.1^2,-1*K.1^6,-1*K.1^2,K.1^6]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(16: Sparse := true);
S := [ K |1,-1,1,-1*K.1^4,K.1^4,-1,-1*K.1^6,K.1^2,-1*K.1^2,K.1^6,-1*K.1^4,K.1^4,-1*K.1^7,K.1^7,K.1^5,-1*K.1^3,K.1^3,-1*K.1^5,K.1,-1*K.1,-1*K.1^6,K.1^2,K.1^6,-1*K.1^2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(16: Sparse := true);
S := [ K |1,-1,1,K.1^4,-1*K.1^4,-1,K.1^2,-1*K.1^6,K.1^6,-1*K.1^2,K.1^4,-1*K.1^4,K.1,-1*K.1,-1*K.1^3,K.1^5,-1*K.1^5,K.1^3,-1*K.1^7,K.1^7,K.1^2,-1*K.1^6,-1*K.1^2,K.1^6]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
x := CR!\[2, 2, -1, 2, 2, -1, 2, 2, 2, 2, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 2, -1, 2, 2, -1, -2, -2, -2, -2, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1]; 
x`IsCharacter := true;
x`Schur := -1;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |2,2,-1,-2,-2,-1,-2*K.1,2*K.1,2*K.1,-2*K.1,1,1,0,0,0,0,0,0,0,0,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,-1,-2,-2,-1,2*K.1,-2*K.1,-2*K.1,2*K.1,1,1,0,0,0,0,0,0,0,0,-1*K.1,K.1,-1*K.1,K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(8: Sparse := true);
S := [ K |2,-2,-1,-2*K.1^2,2*K.1^2,1,2*K.1^3,-2*K.1,2*K.1,-2*K.1^3,K.1^2,-1*K.1^2,0,0,0,0,0,0,0,0,-1*K.1^3,K.1,K.1^3,-1*K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(8: Sparse := true);
S := [ K |2,-2,-1,2*K.1^2,-2*K.1^2,1,-2*K.1,2*K.1^3,-2*K.1^3,2*K.1,-1*K.1^2,K.1^2,0,0,0,0,0,0,0,0,K.1,-1*K.1^3,-1*K.1,K.1^3]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(8: Sparse := true);
S := [ K |2,-2,-1,-2*K.1^2,2*K.1^2,1,-2*K.1^3,2*K.1,-2*K.1,2*K.1^3,K.1^2,-1*K.1^2,0,0,0,0,0,0,0,0,K.1^3,-1*K.1,-1*K.1^3,K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(8: Sparse := true);
S := [ K |2,-2,-1,2*K.1^2,-2*K.1^2,1,2*K.1,-2*K.1^3,2*K.1^3,-2*K.1,-1*K.1^2,K.1^2,0,0,0,0,0,0,0,0,-1*K.1,K.1^3,K.1,-1*K.1^3]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
_ := CharacterTable(G : Check := 0); 
chartbl_48_1:= KnownIrreducibles(CR);