/* Group 1296.2891 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([8, -2, -3, -2, 2, -2, -3, 3, -3, 65, 1010, 226, 114, 1155, 107, 211, 91, 23045, 8469, 2333, 1957, 16134, 17486, 3158, 2494, 710, 718, 55303]); a,b,c,d,e,f,g := Explode([GPC.1, GPC.2, GPC.3, GPC.4, GPC.6, GPC.7, GPC.8]); AssignNames(~GPC, ["a", "b", "c", "d", "d2", "e", "f", "g"]);
GPerm := PermutationGroup< 27 | (2,3)(4,5)(7,9)(10,23)(11,22)(12,24)(13,27)(14,26)(15,25)(16,19)(17,21)(18,20), (1,3,2)(4,6,5)(7,9,8)(10,15,17)(11,13,18)(12,14,16)(19,26,24)(20,27,22)(21,25,23), (4,12,9,19)(5,10,7,20)(6,11,8,21)(13,16,25,24)(14,17,26,22)(15,18,27,23), (4,15,9,27)(5,13,7,25)(6,14,8,26)(10,24,20,16)(11,22,21,17)(12,23,19,18), (4,9)(5,7)(6,8)(10,20)(11,21)(12,19)(13,25)(14,26)(15,27)(16,24)(17,22)(18,23), (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), (1,7,4)(2,8,5)(3,9,6)(10,17,15)(11,18,13)(12,16,14)(19,27,23)(20,25,24)(21,26,22), (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) >;
GLZ := MatrixGroup< 6, Integers() | [[0, 1, 1, 0, 0, -1, 1, 0, 1, 0, 1, -1, -1, 1, 0, 0, -1, 1, 0, 0, -1, 0, 0, 0, 1, -1, 0, 0, 0, 0, 0, 0, 0, -1, -1, 0], [1, 0, 0, 0, 1, -1, 1, 0, 0, 1, 1, 0, 1, -1, 0, -1, 0, -1, -1, 0, -1, -1, -1, 1, 0, 0, 1, 1, 1, 0, 1, -1, 0, 0, 1, -1]] >;
GLFp := MatrixGroup< 4, GF(3) | [[2, 2, 2, 0, 2, 1, 0, 2, 2, 2, 0, 1, 2, 1, 1, 1], [0, 1, 2, 1, 2, 1, 2, 1, 2, 0, 1, 2, 1, 2, 1, 0], [2, 2, 0, 1, 0, 2, 0, 0, 1, 2, 2, 2, 2, 1, 0, 0], [2, 0, 2, 0, 2, 1, 0, 2, 1, 0, 0, 0, 2, 0, 1, 1], [2, 2, 0, 1, 0, 2, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1], [0, 2, 1, 0, 1, 0, 2, 0, 2, 0, 1, 2, 1, 1, 2, 1], [2, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0], [0, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 0, 1, 1, 1, 0]] >;

 
/* Booleans */
RF := recformat< Agroup, Zgroup, abelian, almost_simple, cyclic, metabelian, metacyclic, monomial, nilpotent, perfect, quasisimple, rational, solvable, supersolvable  : BoolElt >; 
booleans_1296_2891 := rec< RF |  
Agroup := false,
Zgroup := false,
abelian := false,
almost_simple := false,
cyclic := false,
metabelian := false,
metacyclic := false,
monomial := false,
nilpotent := false,
perfect := false,
quasisimple := false,
rational := false,
solvable := true,
supersolvable := false>; 

 
/* Character Table */
G:= GLFp;
C := SequenceToConjugacyClasses([car<Integers(), Integers(), G> |< 1, 1, Matrix(4, [1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1])>,< 2, 9, Matrix(4, [1, 2, 0, 0, 0, 2, 0, 0, 0, 2, 2, 1, 0, 1, 0, 1])>,< 2, 108, Matrix(4, [1, 0, 1, 1, 0, 1, 2, 2, 2, 0, 0, 1, 1, 0, 2, 1])>,< 3, 2, Matrix(4, [2, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 2, 0, 0, 0])>,< 3, 24, Matrix(4, [1, 2, 1, 1, 2, 0, 1, 0, 2, 1, 0, 1, 0, 1, 2, 0])>,< 3, 24, Matrix(4, [0, 1, 2, 1, 0, 0, 1, 1, 2, 0, 1, 2, 1, 2, 1, 0])>,< 3, 24, Matrix(4, [1, 0, 0, 0, 2, 0, 2, 1, 1, 1, 2, 2, 0, 0, 0, 1])>,< 3, 24, Matrix(4, [2, 2, 2, 0, 2, 1, 0, 2, 2, 2, 0, 1, 2, 1, 1, 1])>,< 4, 54, Matrix(4, [1, 2, 2, 2, 0, 2, 1, 1, 0, 0, 0, 2, 0, 1, 1, 2])>,< 6, 18, Matrix(4, [0, 2, 0, 2, 0, 2, 0, 0, 2, 2, 2, 0, 1, 1, 0, 2])>,< 6, 72, Matrix(4, [0, 2, 2, 1, 1, 1, 1, 2, 2, 1, 2, 0, 1, 1, 1, 0])>,< 6, 72, Matrix(4, [1, 1, 1, 1, 1, 1, 1, 2, 1, 0, 1, 1, 0, 2, 2, 0])>,< 6, 72, Matrix(4, [2, 1, 1, 2, 1, 1, 2, 0, 1, 2, 1, 1, 2, 2, 2, 2])>,< 6, 216, Matrix(4, [2, 1, 2, 0, 2, 1, 2, 1, 1, 1, 1, 1, 0, 2, 1, 2])>,< 8, 162, Matrix(4, [0, 2, 0, 2, 2, 2, 1, 0, 2, 0, 0, 1, 2, 1, 0, 0])>,< 8, 162, Matrix(4, [1, 0, 1, 1, 1, 0, 1, 2, 2, 1, 2, 0, 1, 0, 2, 1])>,< 9, 144, Matrix(4, [0, 1, 2, 1, 2, 1, 2, 1, 1, 1, 0, 0, 1, 2, 1, 0])>,< 12, 108, Matrix(4, [2, 0, 1, 2, 0, 1, 2, 2, 1, 2, 0, 0, 2, 0, 2, 2])>]); 
CR := CharacterRing(G);
x := CR!\[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]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[2, 2, 0, 2, -1, -1, -1, 2, 2, 2, -1, -1, -1, 0, 0, 0, -1, 2]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
K := CyclotomicField(8: Sparse := true);
S := [ K |2,-2,0,2,-1,-1,-1,2,0,-2,1,1,1,0,-1*K.1-K.1^3,K.1+K.1^3,-1,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
K := CyclotomicField(8: Sparse := true);
S := [ K |2,-2,0,2,-1,-1,-1,2,0,-2,1,1,1,0,K.1+K.1^3,-1*K.1-K.1^3,-1,0]; 
x := CR!S; 
x`IsCharacter := true;
x`Schur := 0;
x`IsIrreducible := true; 
x := CR!\[3, 3, 1, 3, 0, 0, 0, 3, -1, 3, 0, 0, 0, 1, -1, -1, 0, -1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[3, 3, -1, 3, 0, 0, 0, 3, -1, 3, 0, 0, 0, -1, 1, 1, 0, -1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[4, -4, 0, 4, 1, 1, 1, 4, 0, -4, -1, -1, -1, 0, 0, 0, 1, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[6, -2, 0, -3, -3, 3, 0, 0, 2, 1, 1, -2, 1, 0, 0, 0, 0, -1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[6, -2, 0, -3, 0, -3, 3, 0, 2, 1, 1, 1, -2, 0, 0, 0, 0, -1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[6, -2, 0, -3, 3, 0, -3, 0, 2, 1, -2, 1, 1, 0, 0, 0, 0, -1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[8, 0, 2, 8, 2, 2, 2, -1, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[8, 0, -2, 8, 2, 2, 2, -1, 0, 0, 0, 0, 0, 1, 0, 0, -1, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[12, 4, 0, -6, -3, 0, 3, 0, 0, -2, -2, 1, 1, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[12, 4, 0, -6, 0, 3, -3, 0, 0, -2, 1, 1, -2, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[12, 4, 0, -6, 3, -3, 0, 0, 0, -2, 1, -2, 1, 0, 0, 0, 0, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[16, 0, 0, 16, -2, -2, -2, -2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
x := CR!\[18, -6, 0, -9, 0, 0, 0, 0, -2, 3, 0, 0, 0, 0, 0, 0, 0, 1]; 
x`IsCharacter := true;
x`Schur := 1;
x`IsIrreducible := true; 
_ := CharacterTable(G : Check := 0); 
chartbl_1296_2891:= KnownIrreducibles(CR);