# Group 110.1 downloaded from the LMFDB on 21 September 2026. 
 
## Various presentations of this group are stored in this file: 
#	 GPC is polycyclic presentation, GPerm is permutation group 
#	 GLZ, GLFp, GLZN, 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(1570297039,110); a := GPC.1; b := GPC.3;
GPerm := Group( (2,11)(3,10)(4,9)(5,8)(6,7), (2,5,6,10,4)(3,9,11,8,7), (1,11,10,9,8,7,6,5,4,3,2) );
GLFp := Group([[[ Z(11)^0, Z(11)^0 ], [ 0*Z(11), Z(11)^0 ]], [[ Z(11)^5, 0*Z(11) ], [ 0*Z(11), Z(11)^6 ]]]);

 
# Booleans 
booleans_110_1 := rec( 
Agroup := true,
Zgroup := true,
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_110_1:=rec(); 
chartbl_110_1.IsFinite:= true; 
chartbl_110_1.UnderlyingCharacteristic:= 0; 
chartbl_110_1.UnderlyingGroup:= GPC;
chartbl_110_1.Size:= 110;
chartbl_110_1.InfoText:= "Character table for group 110.1 downloaded from the LMFDB."; 
chartbl_110_1.Identifier:= " F11 "; 
chartbl_110_1.NrConjugacyClasses:= 11; 
chartbl_110_1.ConjugacyClasses:= [<identity> of ..., f1*f2^2, f2, f2^4, f2^2, f2^3, f1, f1*f2^4, f1*f2, f1*f2^3, f3];
chartbl_110_1.IdentificationOfConjugacyClasses:= [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11];
chartbl_110_1.ComputedPowerMaps:= [];
chartbl_110_1.ComputedPowerMaps[2]:= [1, 1, 5, 6, 4, 3, 3, 4, 6, 5, 11];
chartbl_110_1.ComputedPowerMaps[5]:= [1, 2, 1, 1, 1, 1, 2, 2, 2, 2, 11];
chartbl_110_1.ComputedPowerMaps[11]:= [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 1];
chartbl_110_1.SizesCentralizers:= [110, 10, 10, 10, 10, 10, 10, 10, 10, 10, 11];
chartbl_110_1.ClassNames:= ["1A", "2A", "5A1", "5A-1", "5A2", "5A-2", "10A1", "10A-1", "10A3", "10A-3", "11A"];
chartbl_110_1.OrderClassRepresentatives:= [1, 2, 5, 5, 5, 5, 10, 10, 10, 10, 11];
chartbl_110_1.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, E(5)^-2, E(5)^2, E(5)^-1, E(5), E(5)^-2, E(5), E(5)^2, E(5)^-1, 1], [1, 1, E(5)^2, E(5)^-2, E(5), E(5)^-1, E(5)^2, E(5)^-1, E(5)^-2, E(5), 1], [1, 1, E(5)^-1, E(5), E(5)^2, E(5)^-2, E(5)^-1, E(5)^-2, E(5), E(5)^2, 1], [1, 1, E(5), E(5)^-1, E(5)^-2, E(5)^2, E(5), E(5)^2, E(5)^-1, E(5)^-2, 1], [1, -1, E(5)^-2, E(5)^2, E(5)^-1, E(5), -1*E(5)^-2, -1*E(5), -1*E(5)^2, -1*E(5)^-1, 1], [1, -1, E(5)^2, E(5)^-2, E(5), E(5)^-1, -1*E(5)^2, -1*E(5)^-1, -1*E(5)^-2, -1*E(5), 1], [1, -1, E(5)^-1, E(5), E(5)^2, E(5)^-2, -1*E(5)^-1, -1*E(5)^-2, -1*E(5), -1*E(5)^2, 1], [1, -1, E(5), E(5)^-1, E(5)^-2, E(5)^2, -1*E(5), -1*E(5)^2, -1*E(5)^-1, -1*E(5)^-2, 1], [10, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1]];
ConvertToLibraryCharacterTableNC(chartbl_110_1);