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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
[31,1,1Δ
≅[[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
= [31,1,1Δ
≅[[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)
Retrieved from " https://en.wikipedia.org/?title=Template:3-sphere_symmetry_groups2&oldid=1008559883"
From Wikipedia, the free encyclopedia
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
[31,1,1Δ
≅[[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
= [31,1,1Δ
≅[[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)
Retrieved from " https://en.wikipedia.org/?title=Template:3-sphere_symmetry_groups2&oldid=1008559883"

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