Eigenschaften der Gruppe G22 mit Hilfe von GAP

Permutationsdarstellung der G22

	pgr22:=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.3*d.1*d.2*d.3*d.2*d.1*d.3*d.2,d.1*d.2*d.3*d.1*d.2*d.1*d.3*d.2*d.1*d.3];
	sg:=Subgroup(g,[g.1]);
	p:=Image(ActionHomomorphism(g,RightTransversal(g,sg),OnRight));
	return p;
	end;
	

Ergebnisse

	G22:=pgr22();
	
	gap> Size(G22);
	240
	
	gap> N22:=NormalSubgroups(G22);;
	
	gap> NTSize(N22);
	Groesse des 1. Normalteilers: 1
	Groesse des 2. Normalteilers: 2
	Groesse des 3. Normalteilers: 4
	Groesse des 4. Normalteilers: 120
	Groesse des 5. Normalteilers: 240
	
	gap> Centre(G22);
	<permutation group of size 4 with 1 generators>
	
	gap> Size(Centre(G22));
	4
	
	gap> IsSolvable(G22);
	false
	
	gap> CompositionSeries(G22);
	[ <permutation group of size 240 with 6 generators>,
	  <permutation group of size 4 with 2 generators>,
	  <permutation group of size 2 with 1 generators>, Group(()) ]
	

Charaktertafel

	gap> ct22:=CharacterTable(G22);
	CharacterTable( <permutation group of size 240 with 3 generators> )

	gap> Display(ct22);
	CT1
	
	      2  4  3   2   2   2   2  2   2   2   2  3   2   2   2  2  4  4  4
	      3  1  .   .   .   .   .  1   1   1   .  .   .   .   .  1  1  1  1
	      5  1  .   1   1   1   1  .   .   .   1  .   1   1   1  .  1  1  1
	
	        1a 2a 10a 20a 20b  5a 6a 12a 12b 10b 4a 20c 20d  5b 3a 4b 4c 2b
	     2P 1a 1a  5a 10a 10a  5b 3a  6a  6a  5b 2b 10b 10b  5a 3a 2b 2b 1a
	     3P 1a 2a 10b 20c 20d  5b 2b  4b  4c 10a 4a 20a 20b  5a 1a 4c 4b 2b
	     5P 1a 2a  2b  4b  4c  1a 6a 12a 12b  2b 4a  4c  4b  1a 3a 4b 4c 2b
	     7P 1a 2a 10b 20c 20d  5b 6a 12b 12a 10a 4a 20a 20b  5a 3a 4c 4b 2b
	    11P 1a 2a 10a 20b 20a  5a 6a 12b 12a 10b 4a 20d 20c  5b 3a 4c 4b 2b
	    13P 1a 2a 10b 20d 20c  5b 6a 12a 12b 10a 4a 20b 20a  5a 3a 4b 4c 2b
	    17P 1a 2a 10b 20d 20c  5b 6a 12a 12b 10a 4a 20b 20a  5a 3a 4b 4c 2b
	    19P 1a 2a 10a 20b 20a  5a 6a 12b 12a 10b 4a 20d 20c  5b 3a 4c 4b 2b
	
	X.1      1  1   1   1   1   1  1   1   1   1  1   1   1   1  1  1  1  1
	X.2      1 -1   1  -1  -1   1  1  -1  -1   1  1  -1  -1   1  1 -1 -1  1
	X.3      2  .   A   B  -B -*A  1  -D   D  *A  .  -C   C  -A -1  E -E -2
	X.4      2  .   A  -B   B -*A  1   D  -D  *A  .   C  -C  -A -1 -E  E -2
	X.5      2  .  *A   C  -C  -A  1  -D   D   A  .  -B   B -*A -1  E -E -2
	X.6      2  .  *A  -C   C  -A  1   D  -D   A  .   B  -B -*A -1 -E  E -2
	X.7      3 -1   A  *A  *A  *A  .   .   .  *A -1   A   A   A  .  3  3  3
	X.8      3  1   A -*A -*A  *A  .   .   .  *A -1  -A  -A   A  . -3 -3  3
	X.9      3 -1  *A   A   A   A  .   .   .   A -1  *A  *A  *A  .  3  3  3
	X.10     3  1  *A  -A  -A   A  .   .   .   A -1 -*A -*A  *A  . -3 -3  3
	X.11     4  .  -1  -1  -1  -1  1   1   1  -1  .  -1  -1  -1  1  4  4  4
	X.12     4  .  -1   1   1  -1  1  -1  -1  -1  .   1   1  -1  1 -4 -4  4
	X.13     4  .   1   D  -D  -1 -1   D  -D   1  .  -D   D  -1  1  F -F -4
	X.14     4  .   1  -D   D  -1 -1  -D   D   1  .   D  -D  -1  1 -F  F -4
	X.15     5  1   .   .   .   . -1  -1  -1   .  1   .   .   . -1  5  5  5
	X.16     5 -1   .   .   .   . -1   1   1   .  1   .   .   . -1 -5 -5  5
	X.17     6  .  -1  -D   D   1  .   .   .  -1  .   D  -D   1  .  G -G -6
	X.18     6  .  -1   D  -D   1  .   .   .  -1  .  -D   D   1  . -G  G -6
	
	A = -E(5)-E(5)^4
	  = (1-ER(5))/2 = -b5
	B = -E(20)^13-E(20)^17
	C = -E(20)-E(20)^9
	D = E(4)
	  = ER(-1) = i
	E = -2*E(4)
	  = -2*ER(-1) = -2i
	F = -4*E(4)
	  = -4*ER(-1) = -4i
	G = -6*E(4)
	  = -6*ER(-1) = -6i
	
	gap> quit;