W系列、E系列、DDD系列的有限羣序列

定義W系列的有限羣序列Wmn(m,n)：
n*m*m階羣Wmn(m,n),
Wmn(2,n)=Dih(4n)
W_32=Wmn(4,2)=GAP[32,13]
W_48=Wmn(4,3)=GAP[48,11]
gap> Wmn:=function(m,n) F:=FreeGroup(2);;G:=F/[F.1^(m*n),F.2^m,F.2^(-1)*F.1*F.2*F.1^(1-2*n)];;return G;end;;
gap> for n in [1..20] do if n=8 or n>14 then Print(IdGroup(Wmn(2,n)),IdGroup(Wmn(4,n)),"未知","\n");else Print(IdGroup(Wmn(2,n)),IdGroup(Wmn(4,n)),IdGroup(Wmn(8,n)),"\n");fi;od;
[ 4, 2 ][ 16, 2 ][ 64, 2 ]
[ 8, 3 ][ 32, 13 ][ 128, 100 ]
[ 12, 4 ][ 48, 11 ][ 192, 14 ]
[ 16, 7 ][ 64, 48 ][ 256, 451 ]
[ 20, 4 ][ 80, 11 ][ 320, 11 ]
[ 24, 6 ][ 96, 24 ][ 384, 89 ]
[ 28, 3 ][ 112, 10 ][ 448, 12 ]
[ 32, 18 ][ 128, 156 ]未知
[ 36, 4 ][ 144, 11 ][ 576, 12 ]
[ 40, 6 ][ 160, 24 ][ 640, 86 ]
[ 44, 3 ][ 176, 10 ][ 704, 12 ]
[ 48, 7 ][ 192, 64 ][ 768, 57251 ]
[ 52, 4 ][ 208, 11 ][ 832, 11 ]
[ 56, 5 ][ 224, 23 ][ 896, 85 ]
[ 60, 12 ][ 240, 72 ]未知
[ 64, 52 ][ 256, 534 ]未知
[ 68, 4 ][ 272, 11 ]未知
[ 72, 6 ][ 288, 25 ]未知
[ 76, 3 ][ 304, 10 ]未知
[ 80, 7 ][ 320, 63 ]未知
m,n爲偶數，E(m,n)是一個m*n階羣。
Emn:=function(m,n) local F,G;;F:=FreeGroup(2);;G:=F/[F.1^m,F.2^n,(F.1*F.2)^2,F.1^(-1)*F.2*F.1^(-1)*F.2];;return G;end;;
DDD(m,n,k)=<a,b|a^m=b^k=1,bab^-1=a^n>
gap> DDD:=function(m,n,k) local F,G;F:=FreeGroup(2);;G:=F/[F.1^m,F.2^k,F.1^(-n)*F.2*F.1*F.2^(-1)];;return G;;end;;
GAP[48,5]=DDD(24,5,2)=<a,b|a^24=b^2=1,bab^-1=a^5>
GAP[16,8]=DDD(8,3,2)=QD_16=<a,b|a^8=b^2=e,bab^(-1)=a^3>
GAP[32,13]=W(4,2)=DDD(8,3,4)
GAP[48,11]=W(4,3)=DDD(12,5,4)
GAP[60,7]=DDD(15,2,4)=<a,b|a^15=b^4=1,bab^-1=a^2>