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User:Tomruen/424 subgroups

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Matrices

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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

Subgroups

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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

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Order Group Generators Structure Diagram
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

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Order Group Generators Structure Diagram
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

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Order Group Generators Structure Diagram
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 Diagram
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

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Order Group Generators Structure Diagram
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

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Order Group Generators Structure Diagram
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

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Order Group Generators Structure Diagram
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

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Order Group Type Generators Structure Diagram
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 12[4+,2+,4+] Doublerot {0123} (1:0/1 2:0/1 4:0/2 ) Z4

2

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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 12[4+,2+,4+]+= [2+,2+,2+] Doublerot {01230123} (1:0/1 2:0/1 ) Z2


1

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