/* Group 192.412 downloaded from the LMFDB on 25 July 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([7, -2, -2, -2, -2, -2, -2, -3, 28, 1373, 36, 4259, 570, 80, 3924, 1411, 102, 3372, 124, 3149]); a,b,c := Explode([GPC.1, GPC.2, GPC.4]); AssignNames(~GPC, ["a", "b", "b2", "c", "c2", "c4", "c8"]);
GPerm := PermutationGroup< 23 | (1,2,6,5)(3,7,10,14)(4,11,13,8)(9,12,16,15)(18,19)(20,21,23,22), (1,3)(2,7)(4,12)(5,14)(6,10)(8,9)(11,16)(13,15)(20,22,23,21), (1,4,6,13)(2,8,5,11)(3,9,10,16)(7,15,14,12)(20,22,23,21), (1,5,6,2)(3,7,10,14)(4,8,13,11)(9,12,16,15), (1,6)(2,5)(3,10)(4,13)(7,14)(8,11)(9,16)(12,15)(20,23)(21,22), (1,6)(2,5)(3,10)(4,13)(7,14)(8,11)(9,16)(12,15), (17,18,19) >;

 
/* Booleans */
RF := recformat< Agroup, Zgroup, abelian, almost_simple, cyclic, metabelian, metacyclic, monomial, nilpotent, perfect, quasisimple, rational, solvable, supersolvable  : BoolElt >; 
booleans_192_412 := 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^2>,< 2, 1, b^2*c^12>,< 2, 1, c^12>,< 3, 2, c^8>,< 4, 2, b^2*c^6>,< 4, 2, c^6>,< 4, 4, a*b^2>,< 4, 4, a>,< 4, 4, a*b^2*c^21>,< 4, 4, a*c^9>,< 4, 6, b>,< 4, 6, b^3>,< 4, 6, b*c^2>,< 4, 6, b^3*c^2>,< 4, 12, a*b>,< 4, 12, a*b*c>,< 4, 12, a*b^3>,< 4, 12, a*b^3*c>,< 6, 2, c^4>,< 6, 2, b^2*c^16>,< 6, 2, b^2*c^20>,< 8, 4, c^3>,< 8, 4, b^2*c^3>,< 8, 12, b*c>,< 8, 12, b^3*c>,< 12, 4, b^2*c^2>,< 12, 4, c^2>,< 12, 8, a*c^8>,< 12, 8, a*b^2*c^8>,< 12, 8, a*c>,< 12, 8, a*b^2*c>,< 24, 4, c>,< 24, 4, c^17>,< 24, 4, b^2*c>,< 24, 4, b^2*c^17>]); 
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*K.1,-1*K.1,K.1,K.1,K.1,-1*K.1,K.1,-1*K.1,-1,-1,1,1,-1,1,-1,1,-1,K.1,-1*K.1,-1,1,-1*K.1,K.1,-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,K.1,K.1,-1*K.1,-1*K.1,-1*K.1,K.1,-1*K.1,K.1,-1,-1,1,1,-1,1,-1,1,-1,-1*K.1,K.1,-1,1,K.1,-1*K.1,K.1,-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*K.1,K.1,-1*K.1,K.1,-1*K.1,K.1,-1*K.1,K.1,-1,1,1,-1,-1,1,-1,-1,1,K.1,-1*K.1,-1,1,K.1,-1*K.1,-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,K.1,-1*K.1,K.1,-1*K.1,K.1,-1*K.1,K.1,-1*K.1,-1,1,1,-1,-1,1,-1,-1,1,-1*K.1,K.1,-1,1,-1*K.1,K.1,K.1,-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*K.1,K.1,-1*K.1,K.1,K.1,-1*K.1,K.1,-1*K.1,1,-1,-1,1,-1,1,-1,-1,1,-1*K.1,K.1,-1,1,K.1,-1*K.1,-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,K.1,-1*K.1,K.1,-1*K.1,-1*K.1,K.1,-1*K.1,K.1,1,-1,-1,1,-1,1,-1,-1,1,K.1,-1*K.1,-1,1,-1*K.1,K.1,K.1,-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*K.1,-1*K.1,K.1,K.1,-1*K.1,K.1,-1*K.1,K.1,1,1,-1,-1,-1,1,-1,1,-1,-1*K.1,K.1,-1,1,-1*K.1,K.1,-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,K.1,K.1,-1*K.1,-1*K.1,K.1,-1*K.1,K.1,-1*K.1,1,1,-1,-1,-1,1,-1,1,-1,K.1,-1*K.1,-1,1,K.1,-1*K.1,K.1,-1*K.1,-1,1,1,-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
x := CR!\[2, 2, 2, 2, -1, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, 2, 2, 0, 0, -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, 0, 0, 0, -2, -2, 2, 2, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0, 0, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 2, 2, 2, 2, -2, -2, 0, 0, 0, 0, 2, 2, -2, -2, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0, 0, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 2, 2, 2, -1, 2, 2, -2, -2, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, 2, 2, 0, 0, -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, -1, 2, 2, -2, 2, 2, -2, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -2, -2, 0, 0, -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, -1, 2, 2, 2, -2, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, -1, -2, -2, 0, 0, -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,-2,2,2,-2,0,0,0,0,-2*K.1,2*K.1,2*K.1,-2*K.1,0,0,0,0,-2,2,-2,0,0,0,0,2,-2,0,0,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |2,-2,2,-2,2,2,-2,0,0,0,0,2*K.1,-2*K.1,-2*K.1,2*K.1,0,0,0,0,-2,2,-2,0,0,0,0,2,-2,0,0,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(4: Sparse := true);
S := [ K |2,-2,2,-2,-1,-2,2,-2*K.1,2*K.1,-2*K.1,2*K.1,0,0,0,0,0,0,0,0,1,-1,1,-2,2,0,0,1,-1,-1*K.1,K.1,K.1,-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 |2,-2,2,-2,-1,-2,2,2*K.1,-2*K.1,2*K.1,-2*K.1,0,0,0,0,0,0,0,0,1,-1,1,-2,2,0,0,1,-1,K.1,-1*K.1,-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 |2,-2,2,-2,-1,-2,2,-2*K.1,-2*K.1,2*K.1,2*K.1,0,0,0,0,0,0,0,0,1,-1,1,2,-2,0,0,1,-1,K.1,-1*K.1,K.1,-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 |2,-2,2,-2,-1,-2,2,2*K.1,2*K.1,-2*K.1,-2*K.1,0,0,0,0,0,0,0,0,1,-1,1,2,-2,0,0,1,-1,-1*K.1,K.1,-1*K.1,K.1,1,-1,-1,1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
x := CR!\[4, 4, 4, 4, -2, -4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -2, -2, 0, 0, 0, 0, 2, 2, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, -4, -4, 4, 0, 0, 0, 0, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, -4, -4, 0, 0, 0, 0, 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, -2, 4, -4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -2, 2, 0, 0, 0, 0, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(24: Sparse := true);
S := [ K |4,-4,-4,4,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,-2,0,0,0,0,0,0,0,0,0,0,-1*K.1-K.1^3-K.1^5+2*K.1^7,-1*K.1-K.1^3-K.1^5+2*K.1^7,K.1+K.1^3+K.1^5-2*K.1^7,K.1+K.1^3+K.1^5-2*K.1^7]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := -1;
x`IsIrreducible := true; 
K := CyclotomicField(24: Sparse := true);
S := [ K |4,-4,-4,4,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,2,-2,0,0,0,0,0,0,0,0,0,0,K.1+K.1^3+K.1^5-2*K.1^7,K.1+K.1^3+K.1^5-2*K.1^7,-1*K.1-K.1^3-K.1^5+2*K.1^7,-1*K.1-K.1^3-K.1^5+2*K.1^7]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := -1;
x`IsIrreducible := true; 
K := CyclotomicField(24: Sparse := true);
S := [ K |4,4,-4,-4,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-2,2,2,0,0,0,0,0,0,0,0,0,0,-1*K.1-K.1^3-K.1^5+2*K.1^7,K.1+K.1^3+K.1^5-2*K.1^7,-1*K.1-K.1^3-K.1^5+2*K.1^7,K.1+K.1^3+K.1^5-2*K.1^7]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := -1;
x`IsIrreducible := true; 
K := CyclotomicField(24: Sparse := true);
S := [ K |4,4,-4,-4,-2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,-2,2,2,0,0,0,0,0,0,0,0,0,0,K.1+K.1^3+K.1^5-2*K.1^7,-1*K.1-K.1^3-K.1^5+2*K.1^7,K.1+K.1^3+K.1^5-2*K.1^7,-1*K.1-K.1^3-K.1^5+2*K.1^7]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := -1;
x`IsIrreducible := true; 
_ := CharacterTable(G : Check := 0); 
chartbl_192_412:= KnownIrreducibles(CR);