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Template:3-sphere symmetry groups2

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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)