/* Group 7776.cw downloaded from the LMFDB on 24 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([10, -2, -2, -2, -2, -2, -3, 3, -3, -3, 3, 60921, 51, 59882, 82, 38723, 170014, 77424, 5934, 144, 85455, 28825, 515, 73926, 159056, 76186, 1716, 8026, 184327, 23057, 24527, 11577, 1747, 116658, 136108, 19478, 9768, 691209, 302419, 172829, 64839, 32449]); a,b,c,d,e,f,g := Explode([GPC.1, GPC.2, GPC.5, GPC.7, GPC.8, GPC.9, GPC.10]); AssignNames(~GPC, ["a", "b", "b2", "b4", "c", "c2", "d", "e", "f", "g"]);
GPerm := PermutationGroup< 27 | (1,23)(2,19)(3,27)(4,26)(5,22)(6,21)(7,20)(8,25)(9,24)(10,12)(13,15)(16,18), (1,17,24,5,14,25,7,15,22,6,11,26,9,10,19,8,13,27,3,18,21,4,16,20)(2,12,23), (1,9,6,5,7,8,2,3)(10,22,11,19,12,21,14,24)(13,27,15,20,18,25,16,26)(17,23) >;

 
/* Booleans */
RF := recformat< Agroup, Zgroup, abelian, almost_simple, cyclic, metabelian, metacyclic, monomial, nilpotent, perfect, quasisimple, rational, solvable, supersolvable  : BoolElt >; 
booleans_7776_cw := rec< RF |  
Agroup := false,
Zgroup := false,
abelian := false,
almost_simple := false,
cyclic := false,
metabelian := false,
metacyclic := false,
nilpotent := false,
perfect := false,
quasisimple := false,
rational := false,
solvable := true,
supersolvable := false>; 

 
/* Character Table */
G:= GPC;
C := SequenceToConjugacyClasses([car<Integers(), Integers(), G> |< 1, 1, Id(G)>,< 2, 27, c^3*g^2>,< 2, 27, b^4*c^3*f^2>,< 2, 36, a*c^3*f>,< 2, 81, b^4*c^2*d^2*g^2>,< 2, 108, a*b^2*c^4*d*f>,< 3, 8, f*g^2>,< 3, 18, e*g^2>,< 3, 24, c^2>,< 3, 48, c^2*d^2*f*g^2>,< 3, 144, d^2*e*f*g>,< 4, 162, b^6*d^2*f^2*g>,< 4, 324, a*b^7*e^2*f>,< 4, 486, b^2*c^3*d^2*e^2*g>,< 4, 972, a*b^7*c^3*f*g>,< 6, 72, a*c^3*g>,< 6, 162, b^4*c^2*d^2*e^2>,< 6, 216, c*g^2>,< 6, 216, a*b^2*c*d^2*f*g^2>,< 6, 216, b^4*c^3>,< 6, 216, a*b^6*c^5*d*e^2*f*g^2>,< 6, 216, a*b^2*c^4*d^2*e^2*f^2*g>,< 6, 216, a*b^6*d^2*e*f^2>,< 6, 432, a*b^2*c^2*d^2*g>,< 6, 432, a*b^4*c*e^2*f*g^2>,< 8, 162, b^7*c*d^2*e*f^2*g^2>,< 8, 162, b*c^3*d*e*f^2*g^2>,< 8, 486, b*c^4*d*e>,< 8, 486, b^7*d*e*f^2*g>,< 12, 324, b^2*c^2*e*f>,< 12, 648, a*b^3*c^4*d*f^2>,< 24, 324, b^5*c^5*d*f^2*g^2>,< 24, 324, b^3*c*e^2*f^2>]); 
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]; 
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]; 
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]; 
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]; 
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]; 
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]; 
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]; 
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]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 0, 0, 2, 2, 0, 2, -1, 2, 2, -1, 2, 2, 0, 0, 2, -1, 0, 2, 0, -1, 0, 0, 0, -1, 2, 2, 0, 0, -1, -1, -1, -1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, -2, -2, 0, 2, 0, 2, 2, 2, 2, 2, -2, 0, 2, 0, 0, 2, -2, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 2, 2, 0, 2, 0, 2, 2, 2, 2, 2, -2, 0, -2, 0, 0, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 0, 0, -2, 2, 0, 2, -1, 2, 2, -1, 2, -2, 0, 0, -2, -1, 0, -2, 0, 1, 0, 0, 0, 1, 2, 2, 0, 0, -1, 1, -1, -1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 0, 0, -2, 2, 0, 2, -1, 2, 2, -1, 2, 2, 0, 0, -2, -1, 0, -2, 0, 1, 0, 0, 0, 1, -2, -2, 0, 0, -1, -1, 1, 1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 0, 0, 2, 2, 0, 2, -1, 2, 2, -1, 2, -2, 0, 0, 2, -1, 0, 2, 0, -1, 0, 0, 0, -1, -2, -2, 0, 0, -1, 1, 1, 1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(8: Sparse := true);
S := [ K |2,-2,2,0,-2,0,2,2,2,2,2,0,0,0,0,0,-2,-2,0,2,0,0,0,0,0,-1*K.1-K.1^3,K.1+K.1^3,-1*K.1-K.1^3,K.1+K.1^3,0,0,-1*K.1-K.1^3,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,-2,2,0,-2,0,2,2,2,2,2,0,0,0,0,0,-2,-2,0,2,0,0,0,0,0,K.1+K.1^3,-1*K.1-K.1^3,K.1+K.1^3,-1*K.1-K.1^3,0,0,K.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,-2,0,-2,0,2,2,2,2,2,0,0,0,0,0,-2,2,0,-2,0,0,0,0,0,-1*K.1-K.1^3,K.1+K.1^3,K.1+K.1^3,-1*K.1-K.1^3,0,0,-1*K.1-K.1^3,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,-2,0,-2,0,2,2,2,2,2,0,0,0,0,0,-2,2,0,-2,0,0,0,0,0,K.1+K.1^3,-1*K.1-K.1^3,-1*K.1-K.1^3,K.1+K.1^3,0,0,K.1+K.1^3,-1*K.1-K.1^3]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
x := CR!\[4, 0, 0, 0, 4, 0, 4, -2, 4, 4, -2, -4, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(8: Sparse := true);
S := [ K |4,0,0,0,-4,0,4,-2,4,4,-2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,-2*K.1-2*K.1^3,2*K.1+2*K.1^3,0,0,0,0,K.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 |4,0,0,0,-4,0,4,-2,4,4,-2,0,0,0,0,0,2,0,0,0,0,0,0,0,0,2*K.1+2*K.1^3,-2*K.1-2*K.1^3,0,0,0,0,-1*K.1-K.1^3,K.1+K.1^3]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
x := CR!\[8, 8, 0, 2, 0, 2, 8, 8, -1, -1, -1, 0, 0, 0, 0, 2, 0, -1, -1, 0, 2, -1, 2, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[8, 8, 0, -2, 0, -2, 8, 8, -1, -1, -1, 0, 0, 0, 0, -2, 0, -1, 1, 0, -2, 1, -2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[8, -8, 0, -2, 0, 2, 8, 8, -1, -1, -1, 0, 0, 0, 0, -2, 0, 1, 1, 0, -2, -1, 2, -1, 1, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[8, -8, 0, 2, 0, -2, 8, 8, -1, -1, -1, 0, 0, 0, 0, 2, 0, 1, -1, 0, 2, 1, -2, 1, -1, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[16, 0, 0, 4, 0, 0, 16, -8, -2, -2, 1, 0, 0, 0, 0, 4, 0, 0, -2, 0, -2, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[16, 0, 0, -4, 0, 0, 16, -8, -2, -2, 1, 0, 0, 0, 0, -4, 0, 0, 2, 0, 2, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[24, 0, 8, 6, 0, 2, -3, 0, 6, -3, 0, 0, 0, 0, 0, -3, 0, 0, 0, -1, 0, 2, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[24, 0, -8, -6, 0, 2, -3, 0, 6, -3, 0, 0, 0, 0, 0, 3, 0, 0, 0, 1, 0, 2, -1, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[24, 0, -8, 6, 0, -2, -3, 0, 6, -3, 0, 0, 0, 0, 0, -3, 0, 0, 0, 1, 0, -2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[24, 0, 8, -6, 0, -2, -3, 0, 6, -3, 0, 0, 0, 0, 0, 3, 0, 0, 0, -1, 0, -2, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[48, 0, 0, 0, 0, 4, -6, 0, -6, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, -2, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[48, 0, 0, 0, 0, -4, -6, 0, -6, 3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
_ := CharacterTable(G : Check := 0); 
chartbl_7776_cw:= KnownIrreducibles(CR);