Eigenschaften der Gruppe G23 mit Hilfe von GAP

Permutationsdarstellung der G23

	pgr23:=function()
	local d,g,sg,p;
	d:=FreeGroup("a","b","c");
	g:=d/[d.1^2,d.2^2,d.3^2,d.1*d.2*d.1*d.2*d.1*d.2*d.1*d.2*d.1*d.2,d.2*d.3*d.2*d.3*d.2*d.3,d.1*d.3*d.1*d.3];
	sg:=Subgroup(g,[g.1,g.2]);
	p:=Image(ActionHomomorphism(g,RightTransversal(g,sg),OnRight));
	return p;
	end;
	

Ergebnisse

        
	G23:=pgr23();
	
	gap> Size(G23);
	120
	
	gap> N23:=NormalSubgroups(G23);;
	
	gap> NTSize(N23);
	Groesse des 1. Normalteilers: 1
	Groesse des 2. Normalteilers: 2
	Groesse des 3. Normalteilers: 60
	Groesse des 4. Normalteilers: 120
	
	gap> Centre(G23);
	Group([ ( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7) ])
	
	gap> Size(Centre(G23));
	2
	
	gap> IsSolvable(G23);
	false
	
	gap> CompositionSeries(G23);
	[ Group([ ( 3, 4)( 5, 6)( 7, 8)( 9,10), ( 2, 3)( 4, 5)( 8, 9)(10,11), 
	      ( 1, 2)( 5, 7)( 6, 8)(11,12), ( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 8)
	        ( 6, 7) ]), Group([ ( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7) ]), 
	  Group(()) ]
	

Charaktertafel

      
	gap> ct23:=CharacterTable(G23);
	CharacterTable( Group([ ( 3, 4)( 5, 6)( 7, 8)( 9,10), 
	  ( 2, 3)( 4, 5)( 8, 9)(10,11), ( 1, 2)( 5, 7)( 6, 8)(11,12) ]) )
	
	gap> Display(ct23);
	CT1
	
	      2  3  3  1  1  3  1   1  1   1  3
	      3  1  .  .  .  .  1   .  1   .  1
	      5  1  .  1  1  .  .   1  .   1  1
	
	        1a 2a 5a 5b 2b 3a 10a 6a 10b 2c
	     2P 1a 1a 5b 5a 1a 3a  5a 3a  5b 1a
	     3P 1a 2a 5b 5a 2b 1a 10b 2c 10a 2c
	     5P 1a 2a 1a 1a 2b 3a  2c 6a  2c 2c
	     7P 1a 2a 5b 5a 2b 3a 10b 6a 10a 2c
	
	X.1      1  1  1  1  1  1   1  1   1  1
	X.2      1 -1  1  1  1  1  -1 -1  -1 -1
	X.3      3 -1  A *A -1  .  *A  .   A  3
	X.4      3  1  A *A -1  . -*A  .  -A -3
	X.5      3 -1 *A  A -1  .   A  .  *A  3
	X.6      3  1 *A  A -1  .  -A  . -*A -3
	X.7      4  . -1 -1  .  1  -1  1  -1  4
	X.8      4  . -1 -1  .  1   1 -1   1 -4
	X.9      5  1  .  .  1 -1   . -1   .  5
	X.10     5 -1  .  .  1 -1   .  1   . -5
	
	A = -E(5)-E(5)^4
	  = (1-ER(5))/2 = -b5
	
	gap> quit;