/* Group 24.2 downloaded from the LMFDB on 13 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([4, -2, -2, -2, -3, 8, 21, 34]); a := Explode([GPC.1]); AssignNames(~GPC, ["a", "a2", "a4", "a8"]);
GPerm := PermutationGroup< 11 | (1,8,4,6,2,7,3,5), (9,11,10), (1,4,2,3)(5,8,6,7), (1,2)(3,4)(5,6)(7,8) >;
GLZ := MatrixGroup< 6, Integers() | [[0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, -1, 1]] >;
GLFp := MatrixGroup< 2, GF(5) | [[4, 1, 3, 4]] >;

 
/* Booleans */
RF := recformat< Agroup, Zgroup, abelian, almost_simple, cyclic, metabelian, metacyclic, monomial, nilpotent, perfect, quasisimple, rational, solvable, supersolvable  : BoolElt >; 
booleans_24_2 := rec< RF |  
Agroup := true,
Zgroup := true,
abelian := true,
almost_simple := false,
cyclic := true,
metabelian := true,
metacyclic := true,
monomial := true,
nilpotent := true,
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^12>,< 3, 1, a^8>,< 3, 1, a^16>,< 4, 1, a^6>,< 4, 1, a^18>,< 6, 1, a^4>,< 6, 1, a^20>,< 8, 1, a^3>,< 8, 1, a^21>,< 8, 1, a^9>,< 8, 1, a^15>,< 12, 1, a^2>,< 12, 1, a^22>,< 12, 1, a^10>,< 12, 1, a^14>,< 24, 1, a>,< 24, 1, a^23>,< 24, 1, a^5>,< 24, 1, a^19>,< 24, 1, a^7>,< 24, 1, a^17>,< 24, 1, a^11>,< 24, 1, a^13>]); 
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(3: Sparse := true);
S := [ K |1,1,K.1^-1,K.1,1,1,K.1,K.1^-1,1,1,1,1,K.1^-1,K.1,K.1,K.1^-1,K.1,K.1^-1,K.1^-1,K.1,K.1,K.1^-1,K.1^-1,K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |1,1,K.1,K.1^-1,1,1,K.1^-1,K.1,1,1,1,1,K.1,K.1^-1,K.1^-1,K.1,K.1^-1,K.1,K.1,K.1^-1,K.1^-1,K.1,K.1,K.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*K.1,K.1,K.1,-1*K.1,-1,-1,-1,-1,K.1,-1*K.1,K.1,-1*K.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,K.1,-1*K.1,-1*K.1,K.1,-1,-1,-1,-1,-1*K.1,K.1,-1*K.1,K.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(3: Sparse := true);
S := [ K |1,1,K.1^-1,K.1,1,1,K.1,K.1^-1,-1,-1,-1,-1,K.1^-1,K.1,K.1,K.1^-1,-1*K.1,-1*K.1^-1,-1*K.1^-1,-1*K.1,-1*K.1,-1*K.1^-1,-1*K.1^-1,-1*K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |1,1,K.1,K.1^-1,1,1,K.1^-1,K.1,-1,-1,-1,-1,K.1,K.1^-1,K.1^-1,K.1,-1*K.1^-1,-1*K.1,-1*K.1,-1*K.1^-1,-1*K.1^-1,-1*K.1,-1*K.1,-1*K.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*K.1^2,K.1^2,-1,-1,K.1^3,-1*K.1,K.1,-1*K.1^3,K.1^2,-1*K.1^2,K.1^2,-1*K.1^2,K.1,-1*K.1^3,-1*K.1,K.1^3,-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 |1,-1,1,1,K.1^2,-1*K.1^2,-1,-1,-1*K.1,K.1^3,-1*K.1^3,K.1,-1*K.1^2,K.1^2,-1*K.1^2,K.1^2,-1*K.1^3,K.1,K.1^3,-1*K.1,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 |1,-1,1,1,-1*K.1^2,K.1^2,-1,-1,-1*K.1^3,K.1,-1*K.1,K.1^3,K.1^2,-1*K.1^2,K.1^2,-1*K.1^2,-1*K.1,K.1^3,K.1,-1*K.1^3,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 |1,-1,1,1,K.1^2,-1*K.1^2,-1,-1,K.1,-1*K.1^3,K.1^3,-1*K.1,-1*K.1^2,K.1^2,-1*K.1^2,K.1^2,K.1^3,-1*K.1,-1*K.1^3,K.1,-1*K.1,K.1^3,K.1,-1*K.1^3]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(12: Sparse := true);
S := [ K |1,1,-1*K.1^2,K.1^4,-1,-1,K.1^4,-1*K.1^2,-1*K.1^3,K.1^3,K.1^3,-1*K.1^3,K.1^2,-1*K.1^4,-1*K.1^4,K.1^2,-1*K.1,K.1^5,-1*K.1^5,K.1,K.1,-1*K.1^5,K.1^5,-1*K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(12: Sparse := true);
S := [ K |1,1,K.1^4,-1*K.1^2,-1,-1,-1*K.1^2,K.1^4,K.1^3,-1*K.1^3,-1*K.1^3,K.1^3,-1*K.1^4,K.1^2,K.1^2,-1*K.1^4,K.1^5,-1*K.1,K.1,-1*K.1^5,-1*K.1^5,K.1,-1*K.1,K.1^5]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(12: Sparse := true);
S := [ K |1,1,-1*K.1^2,K.1^4,-1,-1,K.1^4,-1*K.1^2,K.1^3,-1*K.1^3,-1*K.1^3,K.1^3,K.1^2,-1*K.1^4,-1*K.1^4,K.1^2,K.1,-1*K.1^5,K.1^5,-1*K.1,-1*K.1,K.1^5,-1*K.1^5,K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(12: Sparse := true);
S := [ K |1,1,K.1^4,-1*K.1^2,-1,-1,-1*K.1^2,K.1^4,-1*K.1^3,K.1^3,K.1^3,-1*K.1^3,-1*K.1^4,K.1^2,K.1^2,-1*K.1^4,-1*K.1^5,K.1,-1*K.1,K.1^5,K.1^5,-1*K.1,K.1,-1*K.1^5]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(24: Sparse := true);
S := [ K |1,-1,-1*K.1^4,K.1^8,-1*K.1^6,K.1^6,-1*K.1^8,K.1^4,K.1^9,-1*K.1^3,K.1^3,-1*K.1^9,-1*K.1^10,K.1^2,-1*K.1^2,K.1^10,K.1^11,-1*K.1,K.1^7,-1*K.1^5,K.1^5,-1*K.1^7,K.1,-1*K.1^11]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(24: Sparse := true);
S := [ K |1,-1,K.1^8,-1*K.1^4,K.1^6,-1*K.1^6,K.1^4,-1*K.1^8,-1*K.1^3,K.1^9,-1*K.1^9,K.1^3,K.1^2,-1*K.1^10,K.1^10,-1*K.1^2,-1*K.1,K.1^11,-1*K.1^5,K.1^7,-1*K.1^7,K.1^5,-1*K.1^11,K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(24: Sparse := true);
S := [ K |1,-1,-1*K.1^4,K.1^8,-1*K.1^6,K.1^6,-1*K.1^8,K.1^4,-1*K.1^9,K.1^3,-1*K.1^3,K.1^9,-1*K.1^10,K.1^2,-1*K.1^2,K.1^10,-1*K.1^11,K.1,-1*K.1^7,K.1^5,-1*K.1^5,K.1^7,-1*K.1,K.1^11]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(24: Sparse := true);
S := [ K |1,-1,K.1^8,-1*K.1^4,K.1^6,-1*K.1^6,K.1^4,-1*K.1^8,K.1^3,-1*K.1^9,K.1^9,-1*K.1^3,K.1^2,-1*K.1^10,K.1^10,-1*K.1^2,K.1,-1*K.1^11,K.1^5,-1*K.1^7,K.1^7,-1*K.1^5,K.1^11,-1*K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(24: Sparse := true);
S := [ K |1,-1,-1*K.1^4,K.1^8,K.1^6,-1*K.1^6,-1*K.1^8,K.1^4,-1*K.1^3,K.1^9,-1*K.1^9,K.1^3,K.1^10,-1*K.1^2,K.1^2,-1*K.1^10,K.1^5,-1*K.1^7,K.1,-1*K.1^11,K.1^11,-1*K.1,K.1^7,-1*K.1^5]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(24: Sparse := true);
S := [ K |1,-1,K.1^8,-1*K.1^4,-1*K.1^6,K.1^6,K.1^4,-1*K.1^8,K.1^9,-1*K.1^3,K.1^3,-1*K.1^9,-1*K.1^2,K.1^10,-1*K.1^10,K.1^2,-1*K.1^7,K.1^5,-1*K.1^11,K.1,-1*K.1,K.1^11,-1*K.1^5,K.1^7]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(24: Sparse := true);
S := [ K |1,-1,-1*K.1^4,K.1^8,K.1^6,-1*K.1^6,-1*K.1^8,K.1^4,K.1^3,-1*K.1^9,K.1^9,-1*K.1^3,K.1^10,-1*K.1^2,K.1^2,-1*K.1^10,-1*K.1^5,K.1^7,-1*K.1,K.1^11,-1*K.1^11,K.1,-1*K.1^7,K.1^5]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(24: Sparse := true);
S := [ K |1,-1,K.1^8,-1*K.1^4,-1*K.1^6,K.1^6,K.1^4,-1*K.1^8,-1*K.1^9,K.1^3,-1*K.1^3,K.1^9,-1*K.1^2,K.1^10,-1*K.1^10,K.1^2,K.1^7,-1*K.1^5,K.1^11,-1*K.1,K.1,-1*K.1^11,K.1^5,-1*K.1^7]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
_ := CharacterTable(G : Check := 0); 
chartbl_24_2:= KnownIrreducibles(CR);