# Group 16.7 downloaded from the LMFDB on 12 September 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 a record chartbl_n_i where n is the order 
# of the group and i is which group of that order it is. The record is 
# converted to a character table using ConvertToLibraryCharacterTableNC 

 
# Constructions 
GPC := PcGroupCode(2499614,16); a := GPC.1; b := GPC.2;
GPerm := Group( (1,5)(2,6)(3,7)(4,8), (3,4)(5,8)(6,7), (1,4,2,3)(5,7,6,8), (1,2)(3,4)(5,6)(7,8) );
GLZ := Group([[[-1, 1, 1, 0], [0, 0, 0, -1], [0, 1, 1, 1], [0, -1, 0, 0]], [[0, 1, 0, 1], [0, 1, 0, 0], [0, 0, -1, 0], [1, -1, 0, 0]]]);
GLFp := Group([[[ Z(7)^0, 0*Z(7) ], [ 0*Z(7), Z(7)^3 ]], [[ Z(7)^5, Z(7)^4 ], [ Z(7)^3, Z(7)^5 ]]]);

 
# Booleans 
booleans_16_7 := rec( 
Agroup := false,
Zgroup := false,
abelian := false,
almost_simple := false,
cyclic := false,
metabelian := true,
metacyclic := true,
monomial := true,
nilpotent := true,
perfect := false,
quasisimple := false,
rational := false,
solvable := true,
supersolvable := true); 

 
# Character Table 
chartbl_16_7:=rec(); 
chartbl_16_7.IsFinite:= true; 
chartbl_16_7.UnderlyingCharacteristic:= 0; 
chartbl_16_7.UnderlyingGroup:= GPC;
chartbl_16_7.Size:= 16;
chartbl_16_7.InfoText:= "Character table for group 16.7 downloaded from the LMFDB."; 
chartbl_16_7.Identifier:= " D8 "; 
chartbl_16_7.NrConjugacyClasses:= 7; 
chartbl_16_7.ConjugacyClasses:= [<identity> of ..., f4, f1, f1*f2, f3, f2, f2*f4];
chartbl_16_7.IdentificationOfConjugacyClasses:= [1, 2, 3, 4, 5, 6, 7];
chartbl_16_7.ComputedPowerMaps:= [ , [1, 1, 1, 1, 2, 5, 5]];
chartbl_16_7.SizesCentralizers:= [16, 16, 4, 4, 8, 8, 8];
chartbl_16_7.ClassNames:= ["1A", "2A", "2B", "2C", "4A", "8A1", "8A3"];
chartbl_16_7.OrderClassRepresentatives:= [1, 2, 2, 2, 4, 8, 8];
chartbl_16_7.Irr:= [[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], [2, 2, 0, 0, -2, 0, 0], [2, -2, 0, 0, 0, -1*E(8)-E(8)^-1, E(8)+E(8)^-1], [2, -2, 0, 0, 0, E(8)+E(8)^-1, -1*E(8)-E(8)^-1]];
ConvertToLibraryCharacterTableNC(chartbl_16_7);