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Dihedral Group D_4

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The dihedral group D_4 is one of the two non-Abelian groups of the five groups total of group order 8. It is sometimes called the octic group. An example of D_4 is the symmetry group of the square.

CycleGraphD4

The cycle graph of D_4 is shown above. D_4 has cycle index given by

Z(D_4)=1/8x_1^4+1/4x_2x_1^2+3/8x_2^2+1/4x_4.
(1)
DihedralGroupD4Table

Its multiplication table is illustrated above.

D_4 has representation

I=[1 0; 0 1]
(2)
A=[0 -1; 1 0]
(3)
B=[-1 0; 0 -1]
(4)
C=[0 1; -1 0]
(5)
D=[-1 0; 0 1]
(6)
E=[0 1; 1 0]
(7)
F=[1 0; 0 -1]
(8)
G=[0 -1; -1 0].
(9)

Conjugacy classes include {I}, {B}, {A,C}, {D,F}, and {E,G}. There are 10 subgroups of D_4: {I}, {I,B}, {I,D}, {I,E}, {I,F}, {I,G}, {I,A,B,C}, {I,B,D,F}, and {I,B,E,G}, {1,A,B,C,D,E,F,G}. Of these, {1}, {1,B}, {1,A,B,C}, {1,B,D,F}, {1,B,E,G}, and {1,A,B,C,D,E,F,G} are normal

See also

Cyclic Group C8, Dihedral Group, Dihedral Group D3, Dihedral Group D5, Finite Group C2×C2×C2, Finite Group C2×C4

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References

Cotton, F. A. Chemical Applications of Group Theory, 3rd ed. New York: Wiley, 1990.

Cite this as:

Weisstein, Eric W. "Dihedral Group D_4." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DihedralGroupD4.html

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