Template:3-sphere symmetry groups2

Finite groups
[ ]: Symbol Order [1]+ 1.1 [1] = [ ] 2.1 [2]: Symbol Order [1+,2]+ 1.1 [2]+ 2.1 [2] 4.1 [2,2]: Symbol Order [2+,2+]+ = [(2+,2+,2+)] 1.1 [2+,2+] 2.1 [2,2]+ 4.1 [2+,2] 4.1 [2,2] 8.1 [2,2,2]: Symbol Order [(2+,2+,2+,2+)] = [2+,2+,2+]+ 1.1 [2+,2+,2+] 2.1 [2+,2,2+] 4.1 [(2,2)+,2+] 4 [[2+,2+,2+]] 4 [2,2,2]+ 8 [2+,2,2] 8.1 [(2,2)+,2] 8 [[2+,2,2+]] 8.1 [2,2,2] 16.1 [[2,2,2]]+ 16 [[2,2+,2]] 16 [[2,2,2]] 32	[p]: Symbol Order [p]+ p [p] 2p [p,2]: Symbol Order [p,2]+ 2p [p,2] 4p [2p,2+]: Symbol Order [2p,2+] 4p [2p+,2+] 2p [p,2,2]: Symbol Order [p+,2,2+] 2p [(p,2)+,2+] 2p [p,2,2]+ 4p [p,2,2+] 4p [p+,2,2] 4p [(p,2)+,2] 4p [p,2,2] 8p [2p,2+,2]: Symbol Order [2p+,2+,2+]+ p [2p+,2+,2+] 2p [2p+,2+,2] 4p [2p+,(2,2)+] 4p [2p,(2,2)+] 8p [2p,2+,2] 8p	[p,2,q]: Symbol Order [p+,2,q+] pq [p,2,q]+ 2pq [p+,2,q] 2pq [p,2,q] 4pq [(p,2)+,2q]: Symbol Order [(p,2)+,2q+] 2pq [(p,2)+,2q] 4pq [2p,2,2q]: Symbol Order [2p+,2+,2q+]+= [(2p+,2+,2q+,2+)] pq [2p+,2+,2q+] 2pq [2p,2+,2q+] 4pq [((2p,2)+,(2q,2)+)] 4pq [2p,2+,2q] 8pq [[p,2,p]]: Symbol Order [[p+,2,p+]] 2p2 [[p,2,p]]+ 4p2 [[p,2,p]+] 4p2 [[p,2,p]] 8p2 [[2p,2,2p]]: Symbol Order [[(2p+,2+,2p+,2+)]] 2p2 [[2p+,2+,2p+]] 4p2 [[((2p,2)+,(2p,2)+)]] 8p2 [[2p,2+,2p]] 16p2	[3,3,2]: Symbol Order [(3,3)Δ,2,1+] ≅ [2,2]+ 4 [(3,3)Δ,2] ≅ [2,(2,2)+] 8 [(3,3)⅄,2,1+] ≅ [4,2+] 8 [(3,3)+,2,1+] = [3,3]+ 12.5 [(3,3)⅄,2] ≅ [2,4,2+] 16 [3,3,2,1+] = [3,3] 24 [(3,3)+,2] 24.10 [3,3,2]+ 24.10 [3,3,2] 48.36 [4,3,2]: Symbol Order [1+,4,3+,2,1+] = [3,3]+ 12 [3+,4,2+] 24 [(3,4)+,2+] 24 [1+,4,3+,2] = [(3,3)+,2] 24.10 [3+,4,2,1+] = [3+,4] 24.10 [(4,3)+,2,1+] = [4,3]+ 24.15 [1+,4,3,2,1+] = [3,3] 24 [1+,4,(3,2)+] = [3,3,2]+ 24 [3,4,2+] 48 [4,3+,2] 48.22 [4,(3,2)+] 48 [(4,3)+,2] 48.36 [1+,4,3,2] = [3,3,2] 48.36 [4,3,2,1+] = [4,3] 48.36 [4,3,2]+ 48.36 [4,3,2] 96.5 [5,3,2]: Symbol Order [(5,3)+,2,1+] = [5,3]+ 60.13 [5,3,2,1+] = [5,3] 120.2 [(5,3)+,2] 120.2 [5,3,2]+ 120.2 [5,3,2] 240 (nc)	[31,1,1]: Symbol Order [31,1,1]Δ ≅[[4,2+,4]]+ 32 [31,1,1]⅄ 64 [31,1,1]+ 96.1 [31,1,1] 192.2 <[3,31,1]> = [4,3,3] 384.1 [3[31,1,1]] = [3,4,3] 1152.1 [3,3,3]: Symbol Order [3,3,3]+ 60.13 [3,3,3] 120.1 [[3,3,3]]+ 120.2 [[3,3,3]+] 120.1 [[3,3,3]] 240.1 [4,3,3]: Symbol Order [1+,4,(3,3)Δ] = [31,1,1]Δ ≅[[4,2+,4]]+ 32 [4,(3,3)Δ] = [2+,4[2,2,2]+] ≅[[4,2+,4]] 64 [1+,4,(3,3)⅄] = [31,1,1]⅄ 64 [1+,4,(3,3)+] = [31,1,1]+ 96.1 [4,(3,3)⅄] ≅ [[4,2,4]] 128 [1+,4,3,3] = [31,1,1] 192.2 [4,(3,3)+] 192.1 [4,3,3]+ 192.3 [4,3,3] 384.1 [3,4,3]: Symbol Order [3+,4,3+] 288.1 [3,4,3⅄] = [4,3,3] 384.1 [3,4,3]+ 576.2 [3+,4,3] 576.1 [[3+,4,3+]] 576 (nc) [3,4,3] 1152.1 [[3,4,3]]+ 1152 (nc) [[3,4,3]+] 1152 (nc) [[3,4,3]] 2304 (nc) [5,3,3]: Symbol Order [5,3,3]+ 7200 (nc) [5,3,3] 14400 (nc)