/* Group 162.13 downloaded from the LMFDB on 17 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([5, -2, -3, -3, 3, -3, 10, 1802, 862, 3123, 848, 78, 2704]); a,b,c := Explode([GPC.1, GPC.3, GPC.4]); AssignNames(~GPC, ["a", "a2", "b", "c", "c3"]);
GPerm := PermutationGroup< 27 | (2,3)(4,7)(5,9)(6,8)(10,21)(11,20)(12,19)(13,27)(14,26)(15,25)(16,24)(17,23)(18,22), (4,5,6)(7,9,8)(10,13,17)(11,14,18)(12,15,16)(19,25,24)(20,26,22)(21,27,23), (1,21,12,3,20,11,2,19,10)(4,24,15,6,23,14,5,22,13)(7,27,18,9,26,17,8,25,16), (1,7,4)(2,8,5)(3,9,6)(10,16,13)(11,17,14)(12,18,15)(19,25,22)(20,26,23)(21,27,24), (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_162_13 := 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, 27, a^3*b^2*c^2>,< 3, 2, c^3>,< 3, 6, b>,< 3, 9, a^4*b^2*c^6>,< 3, 9, a^2*b>,< 6, 27, a^5*b*c^2>,< 6, 27, a*c^5>,< 9, 6, c>,< 9, 6, c^2>,< 9, 6, c^4>,< 9, 18, a^4*b*c>,< 9, 18, a^2*c^8>]); 
CR := CharacterRing(G);
x := CR!\[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]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |1,1,1,1,K.1^-1,K.1,K.1,K.1^-1,1,1,1,K.1^-1,K.1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |1,1,1,1,K.1,K.1^-1,K.1^-1,K.1,1,1,1,K.1,K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |1,-1,1,1,K.1^-1,K.1,-1*K.1,-1*K.1^-1,1,1,1,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,-1,1,1,K.1,K.1^-1,-1*K.1^-1,-1*K.1,1,1,1,K.1,K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
x := CR!\[2, 0, 2, 2, 2, 2, 0, 0, -1, -1, -1, -1, -1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(3: Sparse := true);
S := [ K |2,0,2,2,2*K.1^-1,2*K.1,0,0,-1,-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 |2,0,2,2,2*K.1,2*K.1^-1,0,0,-1,-1,-1,-1*K.1,-1*K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
x := CR!\[6, 0, 6, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(9: Sparse := true);
S := [ K |6,0,-3,0,0,0,0,0,-1*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+K.1^2-2*K.1^4-K.1^-4,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(9: Sparse := true);
S := [ K |6,0,-3,0,0,0,0,0,2*K.1-2*K.1^2+K.1^4-K.1^-4,-1*K.1+K.1^2-2*K.1^4-K.1^-4,-1*K.1+K.1^2+K.1^4+2*K.1^-4,0,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(9: Sparse := true);
S := [ K |6,0,-3,0,0,0,0,0,-1*K.1+K.1^2-2*K.1^4-K.1^-4,-1*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]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
_ := CharacterTable(G : Check := 0); 
chartbl_162_13:= KnownIrreducibles(CR);