Eigenschaften der Gruppe G4 mit Hilfe von GAP

Permutationsdarstellung der G4

	pgr4:=function()
	local z,g,sg,p;
	z:=FreeGroup("a","b");
	g:=z/[z.1^3,z.2^3,z.1*z.2*z.1*z.2^2*z.1^2*z.2^2];
	sg:=Subgroup(g,[g.1]);
	p:=Image(ActionHomomorphism(g,RightTransversal(g,sg),OnRight));
	return p;
	end;     
	

Ergebnisse

	G4:=pgr4();

	gap> Size(G4);
	24

	gap> N4:=NormalSubgroups(G4);;

	gap> NTSize(N4);
	Groesse des 1. Normalteilers: 1
	Groesse des 2. Normalteilers: 2
	Groesse des 3. Normalteilers: 8
	Groesse des 4. Normalteilers: 24
	
	gap> Centre(G4);
	Group([ ( 1, 7)( 2, 6)( 3, 8)( 4, 5) ])
	
	gap> Size(Centre(G4));
	2
	
	gap> IsSolvable(G4);
	true
	
	gap> CompositionSeries(G4);
	[ Group([ (2,3,5)(4,6,8), (1,4,7,5)(2,3,6,8), (1,8,7,3)(2,4,6,5), 
	      (1,7)(2,6)(3,8)(4,5) ]), 
	  Group([ (1,4,7,5)(2,3,6,8), (1,8,7,3)(2,4,6,5), (1,7)(2,6)(3,8)(4,5) ]), 
	  Group([ (1,8,7,3)(2,4,6,5), (1,7)(2,6)(3,8)(4,5) ]), 
	  Group([ (1,7)(2,6)(3,8)(4,5) ]), Group(()) ]
	

Charaktertafel

	gap> ct4:=CharacterTable(G4);
	CharacterTable( Group([ (2,3,5)(4,6,8), (1,2,4)(5,7,6) ]) )
	
	gap> Display(ct4);
	CT1

	     2  3   1   1  1  2  1  3
	     3  1   1   1  1  .  1  1

	       1a  3a  3b 6a 4a 6b 2a
	    2P 1a  3b  3a 3a 2a 3b 1a
	    3P 1a  1a  1a 2a 4a 2a 2a
	    5P 1a  3b  3a 6b 4a 6a 2a

	X.1     1   1   1  1  1  1  1
	X.2     1   A  /A /A  1  A  1
	X.3     1  /A   A  A  1 /A  1
	X.4     2  -1  -1  1  .  1 -2
	X.5     2 -/A  -A  A  . /A -2
	X.6     2  -A -/A /A  .  A -2
	X.7     3   .   .  . -1  .  3
	
	A = E(3)
	  = (-1+ER(-3))/2 = b3

	gap> quit;