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.

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

Its multiplication table is illustrated above.

👁 D_4
has representation

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.

Referenced on Wolfram|Alpha

Dihedral Group D_4

Cite this as:

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

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