# Group 40.5 downloaded from the LMFDB on 15 June 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 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(1040612736515,40); a := GPC.1; b := GPC.2;
GPerm := Group( (2,3)(4,5)(6,7)(8,9), (6,8,7,9), (6,7)(8,9), (1,2,4,5,3) );
GLZ := Group([[[1, 0, 0, 0, 0, 0], [-1, -1, -1, -1, 0, 0], [0, 0, 0, 1, 0, 0], [0, 0, 1, 0, 0, 0], [0, 0, 0, 0, 1, 0], [0, 0, 0, 0, 0, 1]], [[0, 0, -1, 0, 0, 0], [0, 0, 0, -1, 0, 0], [1, 1, 1, 1, 0, 0], [-1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 1], [0, 0, 0, 0, -1, 0]]]);
GLFp := Group([[[ Z(5)^0, Z(5)^0 ], [ 0*Z(5), Z(5)^0 ]], [[ Z(5), 0*Z(5) ], [ 0*Z(5), Z(5)^3 ]], [[ Z(5)^0, 0*Z(5) ], [ 0*Z(5), Z(5)^2 ]]]);

 
# Booleans 
booleans_40_5 := rec( 
Agroup := true,
Zgroup := false,
abelian := false,
almost_simple := false,
cyclic := false,
metabelian := true,
metacyclic := true,
monomial := true,
nilpotent := false,
perfect := false,
quasisimple := false,
rational := false,
solvable := true,
supersolvable := true); 

 
# Character Table 
chartbl_40_5:=rec(); 
chartbl_40_5.IsFinite:= true; 
chartbl_40_5.UnderlyingCharacteristic:= 0; 
chartbl_40_5.UnderlyingGroup:= GPC;
chartbl_40_5.Size:= 40;
chartbl_40_5.InfoText:= "Character table for group 40.5 downloaded from the LMFDB."; 
chartbl_40_5.Identifier:= " C4*D5 "; 
chartbl_40_5.NrConjugacyClasses:= 16; 
chartbl_40_5.ConjugacyClasses:= [<identity> of ..., f3*f4^2, f1, f1*f3, f2*f4, f2*f3*f4^3, f1*f2, f1*f2*f3, f4, f4^2, f3, f3*f4, f2, f2*f3*f4^2, f2*f3, f2*f4^3];
chartbl_40_5.IdentificationOfConjugacyClasses:= [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16];
chartbl_40_5.ComputedPowerMaps:= [ , [1, 1, 1, 1, 2, 2, 2, 2, 10, 9, 9, 10, 11, 11, 12, 12], [1, 2, 3, 4, 5, 6, 7, 8, 1, 1, 2, 2, 5, 6, 6, 5]];
chartbl_40_5.SizesCentralizers:= [40, 40, 8, 8, 40, 40, 8, 8, 20, 20, 20, 20, 20, 20, 20, 20];
chartbl_40_5.ClassNames:= ["1A", "2A", "2B", "2C", "4A1", "4A-1", "4B1", "4B-1", "5A1", "5A2", "10A1", "10A3", "20A1", "20A-1", "20A3", "20A-3"];
chartbl_40_5.OrderClassRepresentatives:= [1, 2, 2, 2, 4, 4, 4, 4, 5, 5, 10, 10, 20, 20, 20, 20];
chartbl_40_5.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*E(4), E(4), E(4), -1*E(4), 1, 1, -1, -1, -1*E(4), E(4), E(4), -1*E(4)], [1, -1, -1, 1, E(4), -1*E(4), -1*E(4), E(4), 1, 1, -1, -1, E(4), -1*E(4), -1*E(4), E(4)], [1, -1, 1, -1, -1*E(4), E(4), -1*E(4), E(4), 1, 1, -1, -1, -1*E(4), E(4), E(4), -1*E(4)], [1, -1, 1, -1, E(4), -1*E(4), E(4), -1*E(4), 1, 1, -1, -1, E(4), -1*E(4), -1*E(4), E(4)], [2, 2, 0, 0, 2, 2, 0, 0, E(5)^2+E(5)^-2, E(5)+E(5)^-1, E(5)+E(5)^-1, E(5)^2+E(5)^-2, E(5)^2+E(5)^-2, E(5)^2+E(5)^-2, E(5)+E(5)^-1, E(5)+E(5)^-1], [2, 2, 0, 0, 2, 2, 0, 0, E(5)+E(5)^-1, E(5)^2+E(5)^-2, E(5)^2+E(5)^-2, E(5)+E(5)^-1, E(5)+E(5)^-1, E(5)+E(5)^-1, E(5)^2+E(5)^-2, E(5)^2+E(5)^-2], [2, 2, 0, 0, -2, -2, 0, 0, E(5)^2+E(5)^-2, E(5)+E(5)^-1, E(5)+E(5)^-1, E(5)^2+E(5)^-2, -1*E(5)^2-E(5)^-2, -1*E(5)^2-E(5)^-2, -1*E(5)-E(5)^-1, -1*E(5)-E(5)^-1], [2, 2, 0, 0, -2, -2, 0, 0, E(5)+E(5)^-1, E(5)^2+E(5)^-2, E(5)^2+E(5)^-2, E(5)+E(5)^-1, -1*E(5)-E(5)^-1, -1*E(5)-E(5)^-1, -1*E(5)^2-E(5)^-2, -1*E(5)^2-E(5)^-2], [2, -2, 0, 0, -2*E(20)^5, 2*E(20)^5, 0, 0, -1*E(20)^2-E(20)^-2, E(20)^4+E(20)^-4, -1*E(20)^4-E(20)^-4, E(20)^2+E(20)^-2, E(20)^3+E(20)^7, -1*E(20)^3-E(20)^7, E(20)^3-E(20)^5+E(20)^7, -1*E(20)^3+E(20)^5-E(20)^7], [2, -2, 0, 0, 2*E(20)^5, -2*E(20)^5, 0, 0, -1*E(20)^2-E(20)^-2, E(20)^4+E(20)^-4, -1*E(20)^4-E(20)^-4, E(20)^2+E(20)^-2, -1*E(20)^3-E(20)^7, E(20)^3+E(20)^7, -1*E(20)^3+E(20)^5-E(20)^7, E(20)^3-E(20)^5+E(20)^7], [2, -2, 0, 0, -2*E(20)^5, 2*E(20)^5, 0, 0, E(20)^4+E(20)^-4, -1*E(20)^2-E(20)^-2, E(20)^2+E(20)^-2, -1*E(20)^4-E(20)^-4, -1*E(20)^3+E(20)^5-E(20)^7, E(20)^3-E(20)^5+E(20)^7, -1*E(20)^3-E(20)^7, E(20)^3+E(20)^7], [2, -2, 0, 0, 2*E(20)^5, -2*E(20)^5, 0, 0, E(20)^4+E(20)^-4, -1*E(20)^2-E(20)^-2, E(20)^2+E(20)^-2, -1*E(20)^4-E(20)^-4, E(20)^3-E(20)^5+E(20)^7, -1*E(20)^3+E(20)^5-E(20)^7, E(20)^3+E(20)^7, -1*E(20)^3-E(20)^7]];
ConvertToLibraryCharacterTableNC(chartbl_40_5);