Difference between revisions of "Template:3-sphere symmetry groups2"

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Latest revision as of 13:55, 14 October 2020

Finite groups

[ ]: File:CDel node.png
Symbol	Order
[1]⁺	1.1
[1] = [ ]	2.1

[2]: File:CDel node.png File:CDel 2.png File:CDel node.png
Symbol	Order
[1⁺,2]⁺	1.1
[2]⁺	2.1
[2]	4.1

[2,2]: File:CDel node.png File:CDel 2.png File:CDel node.png File:CDel 2.png File:CDel node.png
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]: File:CDel node.png File:CDel 2.png File:CDel node.png File:CDel 2.png File:CDel node.png File:CDel 2.png File:CDel node.png
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]: File:CDel node.png File:CDel p.png File:CDel node.png
Symbol	Order
[p]⁺	p
[p]	2p

[p,2]: File:CDel node.png File:CDel p.png File:CDel node.png File:CDel 2.png File:CDel node.png
Symbol	Order
[p,2]⁺	2p
[p,2]	4p

[2p,2⁺]: File:CDel node.png File:CDel 2x.png File:CDel p.png File:CDel node.png File:CDel 2x.png File:CDel node.png
Symbol	Order
[2p,2⁺]	4p
[2p⁺,2⁺]	2p

[p,2,2]: File:CDel node.png File:CDel p.png File:CDel node.png File:CDel 2.png File:CDel node.png File:CDel 2.png File:CDel node.png
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]: File:CDel node.png File:CDel 2x.png File:CDel p.png File:CDel node.png File:CDel 2.png File:CDel node.png File:CDel 2.png File:CDel node.png
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]: File:CDel node.png File:CDel p.png File:CDel node.png File:CDel 2.png File:CDel node.png File:CDel q.png File:CDel node.png
Symbol	Order
[p⁺,2,q⁺]	pq
[p,2,q]⁺	2pq
[p⁺,2,q]	2pq
[p,2,q]	4pq
Symbol	Order
[(p,2)⁺,2q⁺]	2pq
[(p,2)⁺,2q]	4pq

[2p,2,2q]: File:CDel node.png File:CDel 2x.png File:CDel p.png File:CDel node.png File:CDel 2.png File:CDel node.png File:CDel 2x.png File:CDel q.png File:CDel node.png
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]]: File:CDel node.png File:CDel p.png File:CDel node.png File:CDel 2.png File:CDel node.png File:CDel p.png File:CDel node.png
Symbol	Order
[[p⁺,2,p⁺]]	2p²
[[p,2,p]]⁺	4p²
[[p,2,p]⁺]	4p²
[[p,2,p]]	8p²

[[2p,2,2p]]: File:CDel node.png File:CDel 2x.png File:CDel p.png File:CDel node.png File:CDel 2.png File:CDel node.png File:CDel 2x.png File:CDel p.png File:CDel node.png
Symbol	Order
[[(2p⁺,2⁺,2p⁺,2⁺)]]	2p²
[[2p⁺,2⁺,2p⁺]]	4p²
[[((2p,2)⁺,(2p,2)⁺)]]	8p²
[[2p,2⁺,2p]]	16p²

[3,3,2]: File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel node.png
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]: File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel node.png
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]: File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 2.png File:CDel node.png
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)

[3^1,1,1]: File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel split1.png File:CDel nodes.png
Symbol	Order
[3^1,1,1]^Δ ≅[[4,2⁺,4]]⁺	32
[3^1,1,1]^⅄	64
[3^1,1,1]⁺	96.1
[3^1,1,1]	192.2
<[3,3^1,1]> = [4,3,3]	384.1
[3[3^1,1,1]] = [3,4,3]	1152.1

[3,3,3]: File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png
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]: File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png
Symbol	Order
[1⁺,4,(3,3)^Δ] = [3^1,1,1]^Δ ≅[[4,2⁺,4]]⁺	32
[4,(3,3)^Δ] = [2⁺,4[2,2,2]⁺] ≅[[4,2⁺,4]]	64
[1⁺,4,(3,3)^⅄] = [3^1,1,1]^⅄	64
[1⁺,4,(3,3)⁺] = [3^1,1,1]⁺	96.1
[4,(3,3)^⅄] ≅ [[4,2,4]]	128
[1⁺,4,3,3] = [3^1,1,1]	192.2
[4,(3,3)⁺]	192.1
[4,3,3]⁺	192.3
[4,3,3]	384.1

[3,4,3]: File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 4.png File:CDel node.png File:CDel 3.png File:CDel node.png
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]: File:CDel node.png File:CDel 5.png File:CDel node.png File:CDel 3.png File:CDel node.png File:CDel 3.png File:CDel node.png
Symbol	Order
[5,3,3]⁺	7200 (nc)
[5,3,3]	14400 (nc)

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