/* Group 864.4669 downloaded from the LMFDB on 26 October 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([8, 2, 2, 2, 3, 2, 3, 2, 3, 353, 41, 482, 66, 515, 25924, 4332, 4820, 2308, 116, 13829, 6925, 1173, 25542, 3374, 11110, 4566, 166, 3079, 3087, 9239]); a,b,c,d := Explode([GPC.1, GPC.2, GPC.5, GPC.7]); AssignNames(~GPC, ["a", "b", "b2", "b4", "c", "c2", "d", "d2"]);
GPerm := PermutationGroup< 10 | (4,5)(8,9), (1,2)(3,4,6,5), (3,6)(4,5), (8,10,9), (1,3,6)(2,4,5), (1,3,6)(2,5,4), (7,8)(9,10), (7,9)(8,10) >;

 
/* Booleans */
RF := recformat< Agroup, Zgroup, abelian, almost_simple, cyclic, metabelian, metacyclic, monomial, nilpotent, perfect, quasisimple, rational, solvable, supersolvable  : BoolElt >; 
booleans_864_4669 := 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 := false>; 

 
/* Character Table */
G:= GPerm;
C := SequenceToConjugacyClasses([car<Integers(), Integers(), G> |< 1, 1, Id(G)>,< 2, 3, G!(7,9)(8,10)>,< 2, 9, G!(2,4)(3,6)>,< 2, 27, G!(3,6)(4,5)(7,8)(9,10)>,< 2, 36, G!(4,5)(7,10)>,< 2, 36, G!(1,4)(2,3)(5,6)(8,10)>,< 3, 4, G!(1,6,3)>,< 3, 4, G!(1,3,6)(2,4,5)>,< 3, 8, G!(8,10,9)>,< 3, 32, G!(1,6,3)(7,10,9)>,< 3, 32, G!(1,6,3)(2,4,5)(7,9,8)>,< 4, 18, G!(1,4,6,2)(3,5)>,< 4, 36, G!(2,4)(7,8,9,10)>,< 4, 36, G!(1,4)(2,6)(3,5)(7,9,10,8)>,< 4, 54, G!(1,5)(2,6,4,3)(7,9)(8,10)>,< 6, 12, G!(1,3,6)(7,9)(8,10)>,< 6, 12, G!(1,6,3)(2,5,4)(7,10)(8,9)>,< 6, 72, G!(1,3,6)(4,5)(7,10)>,< 6, 72, G!(1,6)(2,4)(8,9,10)>,< 6, 72, G!(1,2,6,4,3,5)(8,10)>,< 12, 72, G!(1,6,3)(2,4)(7,10,9,8)>,< 12, 72, G!(1,5,6,4,3,2)(7,8,10,9)>,< 12, 72, G!(1,2,6,4)(3,5)(8,10,9)>,< 12, 72, G!(1,2,6,4)(3,5)(8,9,10)>]); 
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]; 
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]; 
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]; 
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]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 2, 2, 2, 0, 0, 2, 2, -1, -1, -1, 2, 0, 0, 2, 2, 2, 0, -1, 0, 0, 0, -1, -1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 2, -2, -2, 0, 0, 2, 2, 2, 2, 2, 0, 0, 0, 0, 2, 2, 0, -2, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 2, 2, 2, 0, 0, 2, 2, -1, -1, -1, -2, 0, 0, -2, 2, 2, 0, -1, 0, 0, 0, 1, 1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(12: Sparse := true);
S := [ K |2,2,-2,-2,0,0,2,2,-1,-1,-1,0,0,0,0,2,2,0,1,0,0,0,-1*K.1-K.1^-1,K.1+K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(12: Sparse := true);
S := [ K |2,2,-2,-2,0,0,2,2,-1,-1,-1,0,0,0,0,2,2,0,1,0,0,0,K.1+K.1^-1,-1*K.1-K.1^-1]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[3, -1, 3, -1, 1, 1, 3, 3, 0, 0, 0, 3, -1, -1, -1, -1, -1, 1, 0, 1, -1, -1, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[3, -1, 3, -1, -1, -1, 3, 3, 0, 0, 0, 3, 1, 1, -1, -1, -1, -1, 0, -1, 1, 1, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[3, -1, 3, -1, -1, 1, 3, 3, 0, 0, 0, -3, 1, -1, 1, -1, -1, -1, 0, 1, -1, 1, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[3, -1, 3, -1, 1, -1, 3, 3, 0, 0, 0, -3, -1, 1, 1, -1, -1, 1, 0, -1, 1, -1, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[4, 4, 0, 0, 0, 2, -2, 1, 4, -2, 1, 0, 0, 2, 0, -2, 1, 0, 0, -1, -1, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[4, 4, 0, 0, 2, 0, 1, -2, 4, 1, -2, 0, 2, 0, 0, 1, -2, -1, 0, 0, 0, -1, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[4, 4, 0, 0, -2, 0, 1, -2, 4, 1, -2, 0, -2, 0, 0, 1, -2, 1, 0, 0, 0, 1, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[4, 4, 0, 0, 0, -2, -2, 1, 4, -2, 1, 0, 0, -2, 0, -2, 1, 0, 0, 1, 1, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[6, -2, -6, 2, 0, 0, 6, 6, 0, 0, 0, 0, 0, 0, 0, -2, -2, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[8, 8, 0, 0, 0, 0, -4, 2, -4, 2, -1, 0, 0, 0, 0, -4, 2, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[8, 8, 0, 0, 0, 0, 2, -4, -4, -1, 2, 0, 0, 0, 0, 2, -4, 0, 0, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[12, -4, 0, 0, 0, 2, -6, 3, 0, 0, 0, 0, 0, -2, 0, 2, -1, 0, 0, -1, 1, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[12, -4, 0, 0, 2, 0, 3, -6, 0, 0, 0, 0, -2, 0, 0, -1, 2, -1, 0, 0, 0, 1, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[12, -4, 0, 0, -2, 0, 3, -6, 0, 0, 0, 0, 2, 0, 0, -1, 2, 1, 0, 0, 0, -1, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[12, -4, 0, 0, 0, -2, -6, 3, 0, 0, 0, 0, 0, 2, 0, 2, -1, 0, 0, 1, -1, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
_ := CharacterTable(G : Check := 0); 
chartbl_864_4669:= KnownIrreducibles(CR);