# Group 16.2 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(10245,16); a := GPC.1; b := GPC.3;
GPerm := Group( (1,4,2,3), (5,8,6,7), (1,2)(3,4), (5,6)(7,8) );
GLZ := Group([[[1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 0, 1], [0, 0, -1, 0]], [[0, 1, 0, 0], [-1, 0, 0, 0], [0, 0, 1, 0], [0, 0, 0, 1]]]);
GLFp := Group([[[ Z(5)^0, 0*Z(5) ], [ 0*Z(5), Z(5) ]], [[ Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^3 ]]]);

 
# Booleans 
booleans_16_2 := rec( 
Agroup := true,
Zgroup := false,
abelian := true,
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_2:=rec(); 
chartbl_16_2.IsFinite:= true; 
chartbl_16_2.UnderlyingCharacteristic:= 0; 
chartbl_16_2.UnderlyingGroup:= GPC;
chartbl_16_2.Size:= 16;
chartbl_16_2.InfoText:= "Character table for group 16.2 downloaded from the LMFDB."; 
chartbl_16_2.Identifier:= " C4^2 "; 
chartbl_16_2.NrConjugacyClasses:= 16; 
chartbl_16_2.ConjugacyClasses:= [<identity> of ..., f4, f2, f2*f4, f3, f3*f4, f1, f1*f2, f2*f3, f2*f3*f4, f1*f4, f1*f2*f4, f1*f3, f1*f2*f3*f4, f1*f3*f4, f1*f2*f3];
chartbl_16_2.IdentificationOfConjugacyClasses:= [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16];
chartbl_16_2.ComputedPowerMaps:= [ , [1, 1, 1, 1, 2, 2, 3, 3, 2, 2, 3, 3, 4, 4, 4, 4]];
chartbl_16_2.SizesCentralizers:= [16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16, 16];
chartbl_16_2.ClassNames:= ["1A", "2A", "2B", "2C", "4A1", "4A-1", "4B1", "4B-1", "4C1", "4C-1", "4D1", "4D-1", "4E1", "4E-1", "4F1", "4F-1"];
chartbl_16_2.OrderClassRepresentatives:= [1, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4];
chartbl_16_2.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, -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, 1, -1, 1], [1, -1, -1, 1, -1, 1, -1*E(4), -1*E(4), -1*E(4), E(4), E(4), -1*E(4), E(4), -1, E(4), 1], [1, -1, -1, 1, -1, 1, E(4), E(4), E(4), -1*E(4), -1*E(4), E(4), -1*E(4), -1, -1*E(4), 1], [1, -1, -1, 1, 1, -1, -1*E(4), -1*E(4), E(4), -1*E(4), E(4), E(4), E(4), 1, -1*E(4), -1], [1, -1, -1, 1, 1, -1, E(4), E(4), -1*E(4), E(4), -1*E(4), -1*E(4), -1*E(4), 1, E(4), -1], [1, -1, 1, -1, -1*E(4), E(4), -1, 1, E(4), -1*E(4), -1, -1*E(4), 1, E(4), E(4), -1*E(4)], [1, -1, 1, -1, E(4), -1*E(4), -1, 1, -1*E(4), E(4), -1, E(4), 1, -1*E(4), -1*E(4), E(4)], [1, -1, 1, -1, -1*E(4), E(4), 1, -1, -1*E(4), E(4), 1, E(4), -1, E(4), -1*E(4), -1*E(4)], [1, -1, 1, -1, E(4), -1*E(4), 1, -1, E(4), -1*E(4), 1, -1*E(4), -1, -1*E(4), E(4), E(4)], [1, 1, -1, -1, -1*E(4), -1*E(4), E(4), -1*E(4), -1, -1, -1*E(4), 1, E(4), E(4), 1, E(4)], [1, 1, -1, -1, E(4), E(4), -1*E(4), E(4), -1, -1, E(4), 1, -1*E(4), -1*E(4), 1, -1*E(4)], [1, 1, -1, -1, -1*E(4), -1*E(4), -1*E(4), E(4), 1, 1, E(4), -1, -1*E(4), E(4), -1, E(4)], [1, 1, -1, -1, E(4), E(4), E(4), -1*E(4), 1, 1, -1*E(4), -1, E(4), -1*E(4), -1, -1*E(4)]];
ConvertToLibraryCharacterTableNC(chartbl_16_2);