/* Group 497.1 downloaded from the LMFDB on 28 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([2, -7, -71, 1261]); a,b := Explode([GPC.1, GPC.2]); AssignNames(~GPC, ["a", "b"]);
GPerm := PermutationGroup< 71 | (2,46,38,33,21,49,31)(3,20,4,65,41,26,61)(5,39,7,58,10,51,50)(6,13,44,19,30,28,9)(8,32,47,12,70,53,69)(11,25,16,37,59,55,17)(14,18,56,62,48,57,36)(15,63,22,23,68,34,66)(24,42,71,27,35,40,52)(29,54,43,45,64,67,60), (1,71,70,69,68,67,66,65,64,63,62,61,60,59,58,57,56,55,54,53,52,51,50,49,48,47,46,45,44,43,42,41,40,39,38,37,36,35,34,33,32,31,30,29,28,27,26,25,24,23,22,21,20,19,18,17,16,15,14,13,12,11,10,9,8,7,6,5,4,3,2) >;
GLFp := MatrixGroup< 2, GF(71) | [[1, 1, 0, 1], [37, 0, 0, 30]] >;

 
/* Booleans */
RF := recformat< Agroup, Zgroup, abelian, almost_simple, cyclic, metabelian, metacyclic, monomial, nilpotent, perfect, quasisimple, rational, solvable, supersolvable  : BoolElt >; 
booleans_497_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)>,< 7, 71, a^2*b^27>,< 7, 71, a^5*b^53>,< 7, 71, a^4*b^32>,< 7, 71, a^3*b^41>,< 7, 71, a^6*b^4>,< 7, 71, a*b^33>,< 71, 7, b>,< 71, 7, b^23>,< 71, 7, b^2>,< 71, 7, b^7>,< 71, 7, b^4>,< 71, 7, b^14>,< 71, 7, b^5>,< 71, 7, b^28>,< 71, 7, b^10>,< 71, 7, b^13>]); 
CR := CharacterRing(G);
x := CR!\[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(7: Sparse := true);
S := [ K |1,K.1^-3,K.1^2,K.1^-1,K.1,K.1^3,K.1^-2,1,1,1,1,1,1,1,1,1,1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(7: Sparse := true);
S := [ K |1,K.1^3,K.1^-2,K.1,K.1^-1,K.1^-3,K.1^2,1,1,1,1,1,1,1,1,1,1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(7: Sparse := true);
S := [ K |1,K.1^-2,K.1^-1,K.1^-3,K.1^3,K.1^2,K.1,1,1,1,1,1,1,1,1,1,1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(7: Sparse := true);
S := [ K |1,K.1^2,K.1,K.1^3,K.1^-3,K.1^-2,K.1^-1,1,1,1,1,1,1,1,1,1,1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(7: Sparse := true);
S := [ K |1,K.1^-1,K.1^3,K.1^2,K.1^-2,K.1,K.1^-3,1,1,1,1,1,1,1,1,1,1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(7: Sparse := true);
S := [ K |1,K.1,K.1^-3,K.1^-2,K.1^2,K.1^-1,K.1^3,1,1,1,1,1,1,1,1,1,1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(71: Sparse := true);
S := [ K |7,0,0,0,0,0,0,K.1^23+K.1^26+K.1^34+K.1^-32+K.1^-30+K.1^-20+K.1^-1,K.1^14+K.1^21+K.1^22+K.1^33+K.1^-9+K.1^-6+K.1^-4,K.1^5+K.1^8+K.1^12+K.1^18+K.1^27+K.1^29+K.1^-28,K.1^13+K.1^17+K.1^35+K.1^-24+K.1^-16+K.1^-15+K.1^-10,K.1+K.1^20+K.1^30+K.1^32+K.1^-34+K.1^-26+K.1^-23,K.1^2+K.1^3+K.1^19+K.1^25+K.1^-31+K.1^-11+K.1^-7,K.1^7+K.1^11+K.1^31+K.1^-25+K.1^-19+K.1^-3+K.1^-2,K.1^28+K.1^-29+K.1^-27+K.1^-18+K.1^-12+K.1^-8+K.1^-5,K.1^10+K.1^15+K.1^16+K.1^24+K.1^-35+K.1^-17+K.1^-13,K.1^4+K.1^6+K.1^9+K.1^-33+K.1^-22+K.1^-21+K.1^-14]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(71: Sparse := true);
S := [ K |7,0,0,0,0,0,0,K.1+K.1^20+K.1^30+K.1^32+K.1^-34+K.1^-26+K.1^-23,K.1^4+K.1^6+K.1^9+K.1^-33+K.1^-22+K.1^-21+K.1^-14,K.1^28+K.1^-29+K.1^-27+K.1^-18+K.1^-12+K.1^-8+K.1^-5,K.1^10+K.1^15+K.1^16+K.1^24+K.1^-35+K.1^-17+K.1^-13,K.1^23+K.1^26+K.1^34+K.1^-32+K.1^-30+K.1^-20+K.1^-1,K.1^7+K.1^11+K.1^31+K.1^-25+K.1^-19+K.1^-3+K.1^-2,K.1^2+K.1^3+K.1^19+K.1^25+K.1^-31+K.1^-11+K.1^-7,K.1^5+K.1^8+K.1^12+K.1^18+K.1^27+K.1^29+K.1^-28,K.1^13+K.1^17+K.1^35+K.1^-24+K.1^-16+K.1^-15+K.1^-10,K.1^14+K.1^21+K.1^22+K.1^33+K.1^-9+K.1^-6+K.1^-4]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(71: Sparse := true);
S := [ K |7,0,0,0,0,0,0,K.1^28+K.1^-29+K.1^-27+K.1^-18+K.1^-12+K.1^-8+K.1^-5,K.1^23+K.1^26+K.1^34+K.1^-32+K.1^-30+K.1^-20+K.1^-1,K.1^2+K.1^3+K.1^19+K.1^25+K.1^-31+K.1^-11+K.1^-7,K.1^14+K.1^21+K.1^22+K.1^33+K.1^-9+K.1^-6+K.1^-4,K.1^5+K.1^8+K.1^12+K.1^18+K.1^27+K.1^29+K.1^-28,K.1^10+K.1^15+K.1^16+K.1^24+K.1^-35+K.1^-17+K.1^-13,K.1^13+K.1^17+K.1^35+K.1^-24+K.1^-16+K.1^-15+K.1^-10,K.1^7+K.1^11+K.1^31+K.1^-25+K.1^-19+K.1^-3+K.1^-2,K.1^4+K.1^6+K.1^9+K.1^-33+K.1^-22+K.1^-21+K.1^-14,K.1+K.1^20+K.1^30+K.1^32+K.1^-34+K.1^-26+K.1^-23]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(71: Sparse := true);
S := [ K |7,0,0,0,0,0,0,K.1^5+K.1^8+K.1^12+K.1^18+K.1^27+K.1^29+K.1^-28,K.1+K.1^20+K.1^30+K.1^32+K.1^-34+K.1^-26+K.1^-23,K.1^7+K.1^11+K.1^31+K.1^-25+K.1^-19+K.1^-3+K.1^-2,K.1^4+K.1^6+K.1^9+K.1^-33+K.1^-22+K.1^-21+K.1^-14,K.1^28+K.1^-29+K.1^-27+K.1^-18+K.1^-12+K.1^-8+K.1^-5,K.1^13+K.1^17+K.1^35+K.1^-24+K.1^-16+K.1^-15+K.1^-10,K.1^10+K.1^15+K.1^16+K.1^24+K.1^-35+K.1^-17+K.1^-13,K.1^2+K.1^3+K.1^19+K.1^25+K.1^-31+K.1^-11+K.1^-7,K.1^14+K.1^21+K.1^22+K.1^33+K.1^-9+K.1^-6+K.1^-4,K.1^23+K.1^26+K.1^34+K.1^-32+K.1^-30+K.1^-20+K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(71: Sparse := true);
S := [ K |7,0,0,0,0,0,0,K.1^13+K.1^17+K.1^35+K.1^-24+K.1^-16+K.1^-15+K.1^-10,K.1^7+K.1^11+K.1^31+K.1^-25+K.1^-19+K.1^-3+K.1^-2,K.1^4+K.1^6+K.1^9+K.1^-33+K.1^-22+K.1^-21+K.1^-14,K.1^28+K.1^-29+K.1^-27+K.1^-18+K.1^-12+K.1^-8+K.1^-5,K.1^10+K.1^15+K.1^16+K.1^24+K.1^-35+K.1^-17+K.1^-13,K.1+K.1^20+K.1^30+K.1^32+K.1^-34+K.1^-26+K.1^-23,K.1^23+K.1^26+K.1^34+K.1^-32+K.1^-30+K.1^-20+K.1^-1,K.1^14+K.1^21+K.1^22+K.1^33+K.1^-9+K.1^-6+K.1^-4,K.1^5+K.1^8+K.1^12+K.1^18+K.1^27+K.1^29+K.1^-28,K.1^2+K.1^3+K.1^19+K.1^25+K.1^-31+K.1^-11+K.1^-7]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(71: Sparse := true);
S := [ K |7,0,0,0,0,0,0,K.1^10+K.1^15+K.1^16+K.1^24+K.1^-35+K.1^-17+K.1^-13,K.1^2+K.1^3+K.1^19+K.1^25+K.1^-31+K.1^-11+K.1^-7,K.1^14+K.1^21+K.1^22+K.1^33+K.1^-9+K.1^-6+K.1^-4,K.1^5+K.1^8+K.1^12+K.1^18+K.1^27+K.1^29+K.1^-28,K.1^13+K.1^17+K.1^35+K.1^-24+K.1^-16+K.1^-15+K.1^-10,K.1^23+K.1^26+K.1^34+K.1^-32+K.1^-30+K.1^-20+K.1^-1,K.1+K.1^20+K.1^30+K.1^32+K.1^-34+K.1^-26+K.1^-23,K.1^4+K.1^6+K.1^9+K.1^-33+K.1^-22+K.1^-21+K.1^-14,K.1^28+K.1^-29+K.1^-27+K.1^-18+K.1^-12+K.1^-8+K.1^-5,K.1^7+K.1^11+K.1^31+K.1^-25+K.1^-19+K.1^-3+K.1^-2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(71: Sparse := true);
S := [ K |7,0,0,0,0,0,0,K.1^4+K.1^6+K.1^9+K.1^-33+K.1^-22+K.1^-21+K.1^-14,K.1^10+K.1^15+K.1^16+K.1^24+K.1^-35+K.1^-17+K.1^-13,K.1^23+K.1^26+K.1^34+K.1^-32+K.1^-30+K.1^-20+K.1^-1,K.1^2+K.1^3+K.1^19+K.1^25+K.1^-31+K.1^-11+K.1^-7,K.1^14+K.1^21+K.1^22+K.1^33+K.1^-9+K.1^-6+K.1^-4,K.1^28+K.1^-29+K.1^-27+K.1^-18+K.1^-12+K.1^-8+K.1^-5,K.1^5+K.1^8+K.1^12+K.1^18+K.1^27+K.1^29+K.1^-28,K.1+K.1^20+K.1^30+K.1^32+K.1^-34+K.1^-26+K.1^-23,K.1^7+K.1^11+K.1^31+K.1^-25+K.1^-19+K.1^-3+K.1^-2,K.1^13+K.1^17+K.1^35+K.1^-24+K.1^-16+K.1^-15+K.1^-10]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(71: Sparse := true);
S := [ K |7,0,0,0,0,0,0,K.1^14+K.1^21+K.1^22+K.1^33+K.1^-9+K.1^-6+K.1^-4,K.1^13+K.1^17+K.1^35+K.1^-24+K.1^-16+K.1^-15+K.1^-10,K.1+K.1^20+K.1^30+K.1^32+K.1^-34+K.1^-26+K.1^-23,K.1^7+K.1^11+K.1^31+K.1^-25+K.1^-19+K.1^-3+K.1^-2,K.1^4+K.1^6+K.1^9+K.1^-33+K.1^-22+K.1^-21+K.1^-14,K.1^5+K.1^8+K.1^12+K.1^18+K.1^27+K.1^29+K.1^-28,K.1^28+K.1^-29+K.1^-27+K.1^-18+K.1^-12+K.1^-8+K.1^-5,K.1^23+K.1^26+K.1^34+K.1^-32+K.1^-30+K.1^-20+K.1^-1,K.1^2+K.1^3+K.1^19+K.1^25+K.1^-31+K.1^-11+K.1^-7,K.1^10+K.1^15+K.1^16+K.1^24+K.1^-35+K.1^-17+K.1^-13]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(71: Sparse := true);
S := [ K |7,0,0,0,0,0,0,K.1^7+K.1^11+K.1^31+K.1^-25+K.1^-19+K.1^-3+K.1^-2,K.1^28+K.1^-29+K.1^-27+K.1^-18+K.1^-12+K.1^-8+K.1^-5,K.1^10+K.1^15+K.1^16+K.1^24+K.1^-35+K.1^-17+K.1^-13,K.1^23+K.1^26+K.1^34+K.1^-32+K.1^-30+K.1^-20+K.1^-1,K.1^2+K.1^3+K.1^19+K.1^25+K.1^-31+K.1^-11+K.1^-7,K.1^4+K.1^6+K.1^9+K.1^-33+K.1^-22+K.1^-21+K.1^-14,K.1^14+K.1^21+K.1^22+K.1^33+K.1^-9+K.1^-6+K.1^-4,K.1^13+K.1^17+K.1^35+K.1^-24+K.1^-16+K.1^-15+K.1^-10,K.1+K.1^20+K.1^30+K.1^32+K.1^-34+K.1^-26+K.1^-23,K.1^5+K.1^8+K.1^12+K.1^18+K.1^27+K.1^29+K.1^-28]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(71: Sparse := true);
S := [ K |7,0,0,0,0,0,0,K.1^2+K.1^3+K.1^19+K.1^25+K.1^-31+K.1^-11+K.1^-7,K.1^5+K.1^8+K.1^12+K.1^18+K.1^27+K.1^29+K.1^-28,K.1^13+K.1^17+K.1^35+K.1^-24+K.1^-16+K.1^-15+K.1^-10,K.1+K.1^20+K.1^30+K.1^32+K.1^-34+K.1^-26+K.1^-23,K.1^7+K.1^11+K.1^31+K.1^-25+K.1^-19+K.1^-3+K.1^-2,K.1^14+K.1^21+K.1^22+K.1^33+K.1^-9+K.1^-6+K.1^-4,K.1^4+K.1^6+K.1^9+K.1^-33+K.1^-22+K.1^-21+K.1^-14,K.1^10+K.1^15+K.1^16+K.1^24+K.1^-35+K.1^-17+K.1^-13,K.1^23+K.1^26+K.1^34+K.1^-32+K.1^-30+K.1^-20+K.1^-1,K.1^28+K.1^-29+K.1^-27+K.1^-18+K.1^-12+K.1^-8+K.1^-5]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
_ := CharacterTable(G : Check := 0); 
chartbl_497_1:= KnownIrreducibles(CR);