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User:Tomruen/3-sphere symmetry groups

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Finite groups

[ ]:
Symbol	Order
[1]⁺	1.1
[1] = [ ]	2.1

[2,1,1]:
Symbol	Order
[2⁺,1,1]⁺	1.1
[2,1,1]⁺	2.1
[2,1,1]	4.1

[2,2,1]:
Symbol	Order
[2⁺,2⁺,1]⁺ = [(2⁺,2⁺,2⁺)]	1.1
[2⁺,2⁺,1]	2.1
[2,2,1]⁺	4.1
[2⁺,2,1]	4.1
[2,2,1]	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,1,1]:
Symbol	Order
[3,1,1]⁺	3.1
[4,1,1]⁺	4.2
[5,1,1]⁺	5.1
[6,1,1]⁺	6.1
[p,1,1]⁺	p
[3,1,1]	6.2
[4,1,1]	8.4
[5,1,1]	10.2
[6,1,1]	12.3
[p,1,1]	2p

[p,2,1]:
Symbol	Order
[3,2,1]⁺	6.1
[4,2,1]⁺	8.3
[5,2,1]⁺	10.2
[6,2,1]⁺	12.3
[p,2,1]⁺	2p
[3,2,1]	12.3
[4,2,1]	16.6
[5,2,1]	20.3
[6,2,1]	24.6
[p,2,1]	4p

[2p,2⁺,1]:
Symbol	Order
[6,2⁺,1]	12.3
[8,2⁺,1]	16.12
[10,2⁺,1]	20.3
[12,2⁺,1]	24.12
[2p,2⁺,1]	4p

[p,2,2]:
Symbol	Order
[3⁺,2,2⁺]	6
[4⁺,2,2⁺]	8
[5⁺,2,2⁺]	10
[6⁺,2,2⁺]	12
[p⁺,2,2⁺]	2p
[(3,2)⁺,2⁺]	6
[(4,2)⁺,2⁺]	8
[(5,2)⁺,2⁺]	10
[(6,2)⁺,2⁺]	12
[(p,2)⁺,2⁺]	2p
[3,2,2]⁺	12
[4,2,2]⁺	16
[5,2,2]⁺	20
[6,2,2]⁺	24
[p,2,2]⁺	4p
[3,2,2⁺]	12
[4,2,2⁺]	16
[5,2,2⁺]	20
[6,2,2⁺]	24
[p,2,2⁺]	4p
[3⁺,2,2]	12
[4⁺,2,2]	16
[5⁺,2,2]	20
[6⁺,2,2]	24
[p⁺,2,2]	4p
[(3,2)⁺,2]	12
[(4,2)⁺,2]	16
[(5,2)⁺,2]	20
[(6,2)⁺,2]	24
[(p,2)⁺,2]	4p
[3,2,2]	24
[4,2,2]	32
[5,2,2]	40
[6,2,2]	48
[p,2,2]	8p

[2p,2⁺,2]:
Symbol	Order
[6⁺,2⁺,2⁺]	6
[8⁺,2⁺,2⁺]	8
[10⁺,2⁺,2⁺]	10
[12⁺,2⁺,2⁺]	12
[2p⁺,2⁺,2⁺]	2p
[6⁺,2⁺,2]	12
[8⁺,2⁺,2]	16
[10⁺,2⁺,2]	20
[12⁺,2⁺,2]	24
[2p⁺,2⁺,2]	4p
[6⁺,(2,2)⁺]	12
[8⁺,(2,2)⁺]	16
[10⁺,(2,2)⁺]	20
[12⁺,(2,2)⁺]	24
[2p⁺,(2,2)⁺]	4p
[6,(2,2)⁺]	24
[8,(2,2)⁺]	32
[10,(2,2)⁺]	40
[12,(2,2)⁺]	48
[2p,(2,2)⁺]	8p
[6,2⁺,2]	24
[8,2⁺,2]	32
[10,2⁺,2]	40
[12,2⁺,2]	48
[2p,2⁺,2]	8p

[p,2,q]:
Symbol	Order
[3⁺,2,3⁺]	9
[4⁺,2,3⁺]	12
[5⁺,2,3⁺]	15
[6⁺,2,3⁺]	18
[4⁺,2,4⁺]	16
[5⁺,2,4⁺]	20
[6⁺,2,4⁺]	24
[5⁺,2,5⁺]	25
[6⁺,2,5⁺]	30
[6⁺,2,6⁺]	36
[p⁺,2,q⁺]	pq
[3,2,3]⁺	18
[4,2,3]⁺	24
[5,2,3]⁺	30
[6,2,3]⁺	36
[4,2,4]⁺	32
[5,2,4]⁺	40
[6,2,4]⁺	48
[5,2,5]⁺	50
[6,2,5]⁺	60
[6,2,6]⁺	72
[p,2,q]⁺	2pq
[3⁺,2,3]	18
[4⁺,2,3]	24
[5⁺,2,3]	30
[6⁺,2,3]	36
[4⁺,2,4]	32
[5⁺,2,4]	40
[6⁺,2,4]	48
[5⁺,2,5]	50
[6⁺,2,5]	60
[6⁺,2,6]	72
[p⁺,2,q]	2pq
[3,2,3]	36
[4,2,3]	48
[5,2,3]	60
[6,2,3]	72
[4,2,4]	64
[5,2,4]	80
[6,2,4]	96
[5,2,5]	100
[6,2,5]	120
[6,2,6]	144
[p,2,q]	4pq

[(p,2)⁺,2q]:
Symbol	Order
[(3,2)⁺,6⁺]	18
[(4,2)⁺,6⁺]	24
[(5,2)⁺,6⁺]	30
[(6,2)⁺,6⁺]	36
[(4,2)⁺,8⁺]	32
[(5,2)⁺,8⁺]	40
[(6,2)⁺,8⁺]	48
[(5,2)⁺,10⁺]	50
[(6,2)⁺,10⁺]	60
[(6,2)⁺,12⁺]	72
[(p,2)⁺,2q⁺]	2pq
[(3,2)⁺,6]	36
[(4,2)⁺,6]	48
[(5,2)⁺,6]	60
[(6,2)⁺,6]	72
[(4,2)⁺,8]	64
[(5,2)⁺,8]	80
[(6,2)⁺,8]	96
[(5,2)⁺,10]	100
[(6,2)⁺,10]	120
[(6,2)⁺,12]	144
[(p,2)⁺,2q]	4pq

[2p,2⁺,2q]:
Symbol	Order
[6⁺,2⁺,6⁺]	18
[8⁺,2⁺,6⁺]	24
[10⁺,2⁺,6⁺]	30
[12⁺,2⁺,6⁺]	36
[8⁺,2⁺,8⁺]	32
[10⁺,2⁺,8⁺]	40
[12⁺,2⁺,8⁺]	48
[10⁺,2⁺,10⁺]	50
[12⁺,2⁺,10⁺]	60
[12⁺,2⁺,12⁺]	72
[2p⁺,2⁺,2q⁺]	2pq
[6,2⁺,6⁺]	36
[8,2⁺,6⁺]	48
[10,2⁺,6⁺]	60
[12,2⁺,6⁺]	72
[8,2⁺,8⁺]	64
[10,2⁺,8⁺]	80
[12,2⁺,8⁺]	96
[10,2⁺,10⁺]	100
[12,2⁺,10⁺]	120
[12,2⁺,12⁺]	144
[2p,2⁺,2q⁺]	4pq
[6,2⁺,6]	72
[8,2⁺,6]	96
[10,2⁺,6]	120
[12,2⁺,6]	144
[8,2⁺,8]	128
[10,2⁺,8]	160
[12,2⁺,8]	192
[10,2⁺,10]	200
[12,2⁺,10]	240
[12,2⁺,12]	288
[2p,2⁺,2q]	8pq

[[p,2,p]]:
Symbol	Order
[[3⁺,2,3⁺]]	18
[[4⁺,2,4⁺]]	32
[[5⁺,2,5⁺]]	50
[[6⁺,2,6⁺]]	72
[[p⁺,2,p⁺]]	2p²
[[3,2,3]]⁺	36
[[4,2,4]]⁺	64
[[5,2,5]]⁺	100
[[6,2,6]]⁺	144
[[p,2,p]]⁺	4p²
[[3,2,3]]	72
[[4,2,4]]	128
[[5,2,5]]	200
[[6,2,6]]	288
[[p,2,p]]	8p²

[[2p,2⁺,2p]]:
Symbol	Order
[[6⁺,2⁺,6⁺]]	36
[[8⁺,2⁺,8⁺]]	64
[[10⁺,2⁺,10⁺]]	100
[[12⁺,2⁺,12⁺]]	144
[[2p⁺,2⁺,2p⁺]]	4p²
[[6,2⁺,6]]	144
[[8,2⁺,8]]	256
[[10,2⁺,10]]	400
[[12,2⁺,12]]	576
[[2p,2⁺,2p]]	16p²

[3,3,2]:
Symbol	Order
[(3,3)⁺,2,1⁺] =[3,3]⁺	12.5
[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
[(4,3)⁺,2,1⁺] =[4,3]⁺	24.15
[3⁺,4,2,1⁺] =[3⁺,4]	24.10
[3⁺,4,2⁺]	24
[(3,4)⁺,2⁺]	24
[4,3,2,1⁺] =[4,3]	48.36
[3,4,2⁺]	48
[4,(3,2)⁺]	48
[(4,3)⁺,2]	48.36
[4,3,2]⁺	48.36
[4,3⁺,2]	48.22
[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)

[3^1,1,1]:
Symbol	Order
[3^1,1,1]⁺ = [1⁺,4,(3,3)⁺]	96.1
[3^1,1,1] = [1⁺,4,3,3]	192.2
<[3,3^1,1]> = [4,3,3]	384.1
[3[3^1,1,1]] = [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)⁺] = [3^1,1,1]⁺	96.1
[1⁺,4,3,3] = [3,3^1,1]	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]⁺	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)

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