User:Tomruen/424 subgroups

Matrices[edit]

The group is order 64, which represent 64 matrices possible from products of the 4 generators, with matrix characteristic structure (1:0/1 2:16/19 4:16/12). It has 1 identity matrix, 16 reflection matrices [], 16 2-fold rotation matrices [2]+, 2 2D central inversions [2+,2+], and one 3D central inversion [2+,2+,2+], 16 order-4 rotoreflective matrices [2+,4+], and 12 rotational 4-fold matrices: 8 [4]+ and 4 double rotations [4+,2+,4+]+.

[4,2,4],
	Reflections				Rotations	Rotoreflection
Name	1	2	3	4	T	Z
Group
Order	2	2	2	2	2	4
Matrix	$\left[{\begin{smallmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}0&-1&0&0\\-1&0&0&0\\0&0&0&1\\0&0&1&0\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}1&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&1\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}0&1&0&0\\1&0&0&0\\0&0&0&-1\\0&0&-1&0\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}0&0&0&1\\0&0&1&0\\0&1&0&0\\1&0&0&0\\\end{smallmatrix}}\right]$	$\left[{\begin{smallmatrix}0&0&0&1\\1&0&0&0\\0&1&0&0\\0&0&1&0\\\end{smallmatrix}}\right]$

Subgroups[edit]

Generators are listed as a set of matrices, index 0 to 3, or letters for extended groups.

The structure is expressed by looking at all possible matrix products of generators. They are counted by order:A/B, where order of matrix M is how many self-products produce the identity. The A count have determinant -1 (reflective), and B count have determinant +1 (pure rotations). Every group has (1:0/1} for the identity matrix.

Extended[edit]


Order	Group	Generators	Structure
128	[[4,2,4]]	{0,1,T}	(1:0/1 2:16/27 4:48/20 8:0/16 )
64	[[4,2,4]⁺]	{0T,T1}	(1:0/1 2:0/19 4:32/12 )
64	[[4,2,4]]⁺	{01,1T1T}	(1:0/1 2:0/27 4:0/20 8:0/16 )
64	[4⁺,2⁺[1⁺,4,2,4,1⁺]] = [4⁺,2⁺[2,2,2]] = [4⁺,2⁺[2^[4]]]	{0,Z}	(1:0/1 2:8/11 4:24/4 8:0/16 )

64 and reflective subgroups[edit]


Order	Group		Generators	Structure
64	[4,2,4]		{0,1,2,3}	(1:0/1 2:16/19 4:16/12 )	D₈×D₈
32	[4,2,4,1⁺] = [4,2,2]	=	{0,1,2,323}	(1:0/1 2:12/11 4:4/4 )	D₈×Z₂²
32	[1⁺,4,2,4] = [2,2,4]	=	{010,1,2,3}	(1:0/1 2:12/11 4:4/4 )	D₈×Z₂²
16	[1⁺,4,2,4,1⁺] = [2,2,2]	=	{010,1,2,323}	(1:0/1 2:8/7 )	D₂⁴
8	[1⁺,4,2,(4,1⁺),1⁺] = [2,2]	=	{010,1,2}		D₂³
8	[4,2,4^*] = [4]	=	{0,1}		D₈
4	[1⁺,4,2,4^*] = [2]	=	{101,1}		D₄
4	[1⁺,(1⁺,4),2,(4,1⁺),1⁺] = [2]	=	{1,2}	(1:0/1 2:2/1 )	D₂²
2	[1⁺(1⁺,4),2,4*] = [ ]		{1}	(1:0/1 2:1/0 )	D₂

32 and rotional subgroups[edit]


Order	Group		Generators	Structure
32	[4,2,4]⁺		{01,12,23}	(1:0/1 2:0/19 4:0/12 )
16	[1⁺,4,2,4]⁺ = [2,2,4]⁺	=	{0101,12,23}	(1:0/1 2:0/11 4:0/4 )
8	[1⁺,1⁺,4,2,4]⁺ = [2,4]⁺		{12,23}
8	[1⁺,4,2⁺,4,1⁺] = [2,2,2]⁺	=	{0101,12,2323}	(1:0/1 2:0/7 )
4	[4⁺,2,4*] = [4]⁺	=	{01}	(1:0/1 2:0/1 4:0/2 )	Z₄
2	[1⁺,4,2,4^*]⁺ = [2]⁺	=	{10}		Z₂
2	[1⁺,(1⁺,4),2⁺,(4,1⁺),1⁺] = [2]⁺		{12}	(1:0/1 2:0/1 )	Z₂


Order	Group		Generators	Structure
16	[(4,2⁺,4,2⁺)] = [4,2⁺,4]⁺		{03,12,0101}	(1:0/1 2:0/5 4:0/2 )
16	[4⁺,2,4⁺]		{01,23}	(1:0/1 2:0/3 4:0/12 )	Z₄²
8	[1⁺,4,1⁺,2,4⁺] = [2⁺,2,4⁺]	=	{0101,23}		Z₂×Z₄
4	[1⁺,4,1⁺,2,4,1⁺] = [2⁺,2,2⁺]	=	{0101,2323}		Z₂²

32[edit]


Order	Group	Generators	Structure
32	[4,(2,4)⁺]	{0,12,23}	(1:0/1 2:4/11 4:12/4 )
32	[(4,2)⁺,4]	{01,12,3}	(1:0/1 2:4/11 4:12/4 )
32	[4,2⁺,4]	{0,12,3}	(1:0/1 2:8/11 4:8/4 )
32	[4⁺,2,4]	{01,2,3}	(1:0/1 2:8/3 4:8/12 )	Z₄×D₈
32	[4,2,4⁺]	{0,1,23}	(1:0/1 2:8/3 4:8/12 )	Z₄×D₈

16[edit]


Order	Group		Generators	Structure
16	[4⁺,2⁺,4]		{012,3}	(1:0/1 2:4/3 4:4/4 )
16	[4,2⁺,4⁺]		{0,123}	(1:0/1 2:4/3 4:4/4 )
16	[4⁺,2,4,1⁺] = [4⁺,2,2]	=	{01,232,3}	(1:0/1 2:4/3 4:4/4 )
16	[1⁺,4,2⁺,4] = [2,(2,2)⁺]	=	{0101,12,3}	(1:0/1 2:4/7 4:4/0 )
16	[((4,2)⁺,(4,2)⁺)]		{012,123}	(1:0/1 2:0/7 4:8/0 )
16	[1⁺,4,1⁺,2,4] = [2⁺,2,4]	=	{0101,2,3}	(1:0/1 2:8/3 4:0/4 )	Z₂×D₈

8[edit]


Order	Group		Generators
8	[4⁺,2⁺,4⁺]		{0123,0132}
8	[1⁺,4,1⁺,2⁺,4] = [2⁺,2⁺,4]	=	{01012,3}
8	[1⁺,4,(2,4)⁺] = [2⁺,(2,4)⁺]	=	{0101,12,23}
8	[1⁺,4,1⁺,2,4,1⁺] = [2⁺,2,2]	=	{0101,2,323}	Z₂³
8	[1⁺,4,2⁺,4,1⁺] = [1⁺,4,2⁺,2] = [(2,2)⁺,2]	=	{0101,12,2323}

4[edit]


Order	Group		Type	Generators	Structure
4	[1⁺,4,1⁺,2,1⁺,4,1⁺] = [2⁺,2,2⁺]	=	Doublerot	{0101,2323}	(1:0/1 2:0/3 )	Z₂²
4	[4⁺,2,4^*] = [4]⁺		Rot	{01}	(1:0/1 2:4/1 4:0/2 )	Z₄
4	1⁄2[4⁺,2⁺,4⁺]		Doublerot	{0123}	(1:0/1 2:0/1 4:0/2 )	Z₄

2[edit]


Order	Group	Type	Count	Generators	Structure
2	[4*,2,4,1+,1+] = []	Ref		{0}	(1:0/1 2:1/0 )	D₂
2	[1⁺,4,2⁺,4*] = [2⁺,2⁺]	Rotoref		{01012}	(1:0/1 2:1/0 )	D₂
2	[1⁺,4⁺,2⁺,4⁺,1⁺] = [2⁺,2⁺,2⁺]	Doublerot	1	{01012323}	(1:0/1 2:0/1 )	Z₂
2	1⁄2[4⁺,2⁺,4⁺]⁺= [2⁺,2⁺,2⁺]	Doublerot		{01230123}	(1:0/1 2:0/1 )	Z₂

1[edit]


Order	Group		Type	Count	Generators	Structure
2	[4,2,4] = []+		Identity	1	{ }	(1:0/1 )	Z₁