/* Group 972.115 downloaded from the LMFDB on 21 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, -3, -3, -3, 3, -3, 141, 36, 170, 12946, 6401, 108, 12611, 6003, 36293, 12108, 4933, 166, 31758, 15889]); a,b,c,d := Explode([GPC.1, GPC.2, GPC.4, GPC.6]); AssignNames(~GPC, ["a", "b", "b2", "c", "c3", "d", "d3"]);
GPerm := PermutationGroup< 27 | (2,3)(4,8)(5,7)(6,9)(10,19)(11,21)(12,20)(13,26)(14,25)(15,27)(16,23)(17,22)(18,24), (2,3)(4,8)(5,7)(6,9)(11,12)(13,17)(14,16)(15,18)(20,21)(22,26)(23,25)(24,27), (10,13,16,12,15,18,11,14,17)(19,26,23,20,27,24,21,25,22), (1,19,10)(2,20,11)(3,21,12)(4,22,13)(5,23,14)(6,24,15)(7,25,16)(8,26,17)(9,27,18), (1,8,5,2,9,6,3,7,4)(10,16,15,11,17,13,12,18,14)(19,26,23,20,27,24,21,25,22), (1,3,2)(4,6,5)(7,9,8)(10,11,12)(13,14,15)(16,17,18), (1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26) >;

 
/* Booleans */
RF := recformat< Agroup, Zgroup, abelian, almost_simple, cyclic, metabelian, metacyclic, monomial, nilpotent, perfect, quasisimple, rational, solvable, supersolvable  : BoolElt >; 
booleans_972_115 := rec< RF |  
Agroup := false,
Zgroup := false,
abelian := false,
almost_simple := false,
cyclic := false,
metabelian := false,
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, 27, a*b^2*c^3*d^5>,< 2, 27, a*b*c^5*d>,< 2, 81, b^3*c^6*d^3>,< 3, 2, d^6>,< 3, 6, c^3>,< 3, 54, b^4*d^6>,< 3, 108, b^2*c^2*d>,< 6, 54, a*b^2*d^2>,< 6, 54, a*b*c^5*d^4>,< 6, 162, b^5*c^6>,< 9, 6, c*d^7>,< 9, 6, c^2*d^5>,< 9, 6, c^4*d>,< 9, 6, c^6*d^5>,< 9, 6, c^3*d>,< 9, 6, c^6*d^2>,< 9, 12, c*d^3>,< 9, 12, c*d>,< 9, 12, c^2*d>,< 18, 54, a*b^2*c^8*d^4>,< 18, 54, a*b^2*c>,< 18, 54, a*b^2*c^2*d^7>,< 18, 54, a*b*c^8*d^8>,< 18, 54, a*b*c^2>,< 18, 54, a*b*c^8*d^5>]); 
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]; 
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]; 
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]; 
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]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 0, 0, 2, 2, 2, -1, -1, 0, 0, -1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 2, 0, 0, 2, 2, 2, -1, 2, 0, 0, -1, -1, 2, 2, -1, 2, -1, -1, -1, -1, -1, -1, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, -2, 0, 0, 2, 2, 2, -1, -2, 0, 0, -1, -1, 2, 2, -1, 2, -1, -1, -1, 1, 1, 1, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 0, 0, -2, 2, 2, -1, -1, 0, 0, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[4, 0, 0, 0, 4, 4, -2, 1, 0, 0, 0, -2, -2, 4, 4, -2, 4, -2, -2, -2, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[6, 0, 2, 0, 6, 6, 0, 0, 0, 2, 0, 0, 0, -3, -3, 0, -3, 0, 0, 0, 0, 0, 0, -1, -1, -1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[6, 0, -2, 0, 6, 6, 0, 0, 0, -2, 0, 0, 0, -3, -3, 0, -3, 0, 0, 0, 0, 0, 0, 1, 1, 1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(9: Sparse := true);
S := [ K |6,2,0,0,6,-3,0,0,-1,0,0,K.1-K.1^2+2*K.1^4+K.1^-4,K.1-K.1^2-K.1^4-2*K.1^-4,0,0,-2*K.1+2*K.1^2-K.1^4+K.1^-4,0,-2*K.1+2*K.1^2-K.1^4+K.1^-4,K.1-K.1^2+2*K.1^4+K.1^-4,K.1-K.1^2-K.1^4-2*K.1^-4,K.1^2+K.1^-2,K.1+K.1^-1,K.1^4+K.1^-4,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(9: Sparse := true);
S := [ K |6,2,0,0,6,-3,0,0,-1,0,0,-2*K.1+2*K.1^2-K.1^4+K.1^-4,K.1-K.1^2+2*K.1^4+K.1^-4,0,0,K.1-K.1^2-K.1^4-2*K.1^-4,0,K.1-K.1^2-K.1^4-2*K.1^-4,-2*K.1+2*K.1^2-K.1^4+K.1^-4,K.1-K.1^2+2*K.1^4+K.1^-4,K.1+K.1^-1,K.1^4+K.1^-4,K.1^2+K.1^-2,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(9: Sparse := true);
S := [ K |6,2,0,0,6,-3,0,0,-1,0,0,K.1-K.1^2-K.1^4-2*K.1^-4,-2*K.1+2*K.1^2-K.1^4+K.1^-4,0,0,K.1-K.1^2+2*K.1^4+K.1^-4,0,K.1-K.1^2+2*K.1^4+K.1^-4,K.1-K.1^2-K.1^4-2*K.1^-4,-2*K.1+2*K.1^2-K.1^4+K.1^-4,K.1^4+K.1^-4,K.1^2+K.1^-2,K.1+K.1^-1,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(9: Sparse := true);
S := [ K |6,0,2,0,-3,0,0,0,0,-1,0,2+2*K.1^4+2*K.1^-4,2+2*K.1-2*K.1^2-2*K.1^-4,K.1-K.1^2-K.1^4-2*K.1^-4,-2*K.1+2*K.1^2-K.1^4+K.1^-4,2-2*K.1+2*K.1^2-2*K.1^4,K.1-K.1^2+2*K.1^4+K.1^-4,-1+K.1-K.1^2+K.1^4,-1-K.1^4-K.1^-4,-1-K.1+K.1^2+K.1^-4,0,0,0,K.1+K.1^-1,K.1^4+K.1^-4,K.1^2+K.1^-2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(9: Sparse := true);
S := [ K |6,0,2,0,-3,0,0,0,0,-1,0,2-2*K.1+2*K.1^2-2*K.1^4,2+2*K.1^4+2*K.1^-4,K.1-K.1^2+2*K.1^4+K.1^-4,K.1-K.1^2-K.1^4-2*K.1^-4,2+2*K.1-2*K.1^2-2*K.1^-4,-2*K.1+2*K.1^2-K.1^4+K.1^-4,-1-K.1+K.1^2+K.1^-4,-1+K.1-K.1^2+K.1^4,-1-K.1^4-K.1^-4,0,0,0,K.1^4+K.1^-4,K.1^2+K.1^-2,K.1+K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(9: Sparse := true);
S := [ K |6,0,2,0,-3,0,0,0,0,-1,0,2+2*K.1-2*K.1^2-2*K.1^-4,2-2*K.1+2*K.1^2-2*K.1^4,-2*K.1+2*K.1^2-K.1^4+K.1^-4,K.1-K.1^2+2*K.1^4+K.1^-4,2+2*K.1^4+2*K.1^-4,K.1-K.1^2-K.1^4-2*K.1^-4,-1-K.1^4-K.1^-4,-1-K.1+K.1^2+K.1^-4,-1+K.1-K.1^2+K.1^4,0,0,0,K.1^2+K.1^-2,K.1+K.1^-1,K.1^4+K.1^-4]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(9: Sparse := true);
S := [ K |6,-2,0,0,6,-3,0,0,1,0,0,K.1-K.1^2+2*K.1^4+K.1^-4,K.1-K.1^2-K.1^4-2*K.1^-4,0,0,-2*K.1+2*K.1^2-K.1^4+K.1^-4,0,-2*K.1+2*K.1^2-K.1^4+K.1^-4,K.1-K.1^2+2*K.1^4+K.1^-4,K.1-K.1^2-K.1^4-2*K.1^-4,-1*K.1^2-K.1^-2,-1*K.1-K.1^-1,-1*K.1^4-K.1^-4,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(9: Sparse := true);
S := [ K |6,-2,0,0,6,-3,0,0,1,0,0,-2*K.1+2*K.1^2-K.1^4+K.1^-4,K.1-K.1^2+2*K.1^4+K.1^-4,0,0,K.1-K.1^2-K.1^4-2*K.1^-4,0,K.1-K.1^2-K.1^4-2*K.1^-4,-2*K.1+2*K.1^2-K.1^4+K.1^-4,K.1-K.1^2+2*K.1^4+K.1^-4,-1*K.1-K.1^-1,-1*K.1^4-K.1^-4,-1*K.1^2-K.1^-2,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(9: Sparse := true);
S := [ K |6,-2,0,0,6,-3,0,0,1,0,0,K.1-K.1^2-K.1^4-2*K.1^-4,-2*K.1+2*K.1^2-K.1^4+K.1^-4,0,0,K.1-K.1^2+2*K.1^4+K.1^-4,0,K.1-K.1^2+2*K.1^4+K.1^-4,K.1-K.1^2-K.1^4-2*K.1^-4,-2*K.1+2*K.1^2-K.1^4+K.1^-4,-1*K.1^4-K.1^-4,-1*K.1^2-K.1^-2,-1*K.1-K.1^-1,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(9: Sparse := true);
S := [ K |6,0,-2,0,-3,0,0,0,0,1,0,2+2*K.1^4+2*K.1^-4,2+2*K.1-2*K.1^2-2*K.1^-4,K.1-K.1^2-K.1^4-2*K.1^-4,-2*K.1+2*K.1^2-K.1^4+K.1^-4,2-2*K.1+2*K.1^2-2*K.1^4,K.1-K.1^2+2*K.1^4+K.1^-4,-1+K.1-K.1^2+K.1^4,-1-K.1^4-K.1^-4,-1-K.1+K.1^2+K.1^-4,0,0,0,-1*K.1-K.1^-1,-1*K.1^4-K.1^-4,-1*K.1^2-K.1^-2]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(9: Sparse := true);
S := [ K |6,0,-2,0,-3,0,0,0,0,1,0,2-2*K.1+2*K.1^2-2*K.1^4,2+2*K.1^4+2*K.1^-4,K.1-K.1^2+2*K.1^4+K.1^-4,K.1-K.1^2-K.1^4-2*K.1^-4,2+2*K.1-2*K.1^2-2*K.1^-4,-2*K.1+2*K.1^2-K.1^4+K.1^-4,-1-K.1+K.1^2+K.1^-4,-1+K.1-K.1^2+K.1^4,-1-K.1^4-K.1^-4,0,0,0,-1*K.1^4-K.1^-4,-1*K.1^2-K.1^-2,-1*K.1-K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(9: Sparse := true);
S := [ K |6,0,-2,0,-3,0,0,0,0,1,0,2+2*K.1-2*K.1^2-2*K.1^-4,2-2*K.1+2*K.1^2-2*K.1^4,-2*K.1+2*K.1^2-K.1^4+K.1^-4,K.1-K.1^2+2*K.1^4+K.1^-4,2+2*K.1^4+2*K.1^-4,K.1-K.1^2-K.1^4-2*K.1^-4,-1-K.1^4-K.1^-4,-1-K.1+K.1^2+K.1^-4,-1+K.1-K.1^2+K.1^4,0,0,0,-1*K.1^2-K.1^-2,-1*K.1-K.1^-1,-1*K.1^4-K.1^-4]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(9: Sparse := true);
S := [ K |12,0,0,0,-6,0,0,0,0,0,0,-2-2*K.1+2*K.1^2+2*K.1^-4,-2+2*K.1-2*K.1^2+2*K.1^4,-4*K.1+4*K.1^2-2*K.1^4+2*K.1^-4,2*K.1-2*K.1^2+4*K.1^4+2*K.1^-4,-2-2*K.1^4-2*K.1^-4,2*K.1-2*K.1^2-2*K.1^4-4*K.1^-4,1+K.1^4+K.1^-4,1+K.1-K.1^2-K.1^-4,1-K.1+K.1^2-K.1^4,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(9: Sparse := true);
S := [ K |12,0,0,0,-6,0,0,0,0,0,0,-2+2*K.1-2*K.1^2+2*K.1^4,-2-2*K.1^4-2*K.1^-4,2*K.1-2*K.1^2+4*K.1^4+2*K.1^-4,2*K.1-2*K.1^2-2*K.1^4-4*K.1^-4,-2-2*K.1+2*K.1^2+2*K.1^-4,-4*K.1+4*K.1^2-2*K.1^4+2*K.1^-4,1+K.1-K.1^2-K.1^-4,1-K.1+K.1^2-K.1^4,1+K.1^4+K.1^-4,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(9: Sparse := true);
S := [ K |12,0,0,0,-6,0,0,0,0,0,0,-2-2*K.1^4-2*K.1^-4,-2-2*K.1+2*K.1^2+2*K.1^-4,2*K.1-2*K.1^2-2*K.1^4-4*K.1^-4,-4*K.1+4*K.1^2-2*K.1^4+2*K.1^-4,-2+2*K.1-2*K.1^2+2*K.1^4,2*K.1-2*K.1^2+4*K.1^4+2*K.1^-4,1-K.1+K.1^2-K.1^4,1+K.1^4+K.1^-4,1+K.1-K.1^2-K.1^-4,0,0,0,0,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
_ := CharacterTable(G : Check := 0); 
chartbl_972_115:= KnownIrreducibles(CR);