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Dihedral Group D_6


The dihedral group D_6 gives the group of symmetries of a regular hexagon. The group generators are given by a counterclockwise rotation through pi/3 radians and reflection in a line joining the midpoints of two opposite edges. If x denotes rotation and y reflection, we have

 D_6=<x,y:x^6=y^2=1,xy=yx^(-1)>.
(1)

From this, the group elements can be listed as

 D_6={x^i,yx^i:0<=i<=5}.
(2)

The conjugacy classes of D_6 are given by

 {1},{x,x^5},{x^2,x^4},{x^3},{y,yx^2,yx^4},{yx,yx^3,yx^5}.
(3)

The set of elements which by themselves make up conjugacy classes are in the center of G, denoted Z(G), so

 Z(D_6)={1,x^3}.
(4)

The commutator subgroup is given by

 D_6^'={1,x^2,x^4},
(5)

which can be used to find the Abelianization. The set D_6/D_6^' of all left cosets of D_6^' is given by

1D_6^'={1,x^2,x^4},xD_6^'={x,x^3,x^5}
(6)
yD_6^'={y,yx^2,yx^4},yxD_6^'={yx,yx^3,yx^5}.
(7)

Thus we appear to have two generators for this group, namely xD_6^' and yD_6^'. Therefore, Abelianization gives D_6/D_6^'=C_2×C_2.

It is also known that D_6=C_2×S_3 where S_3 is the symmetric group. Furthermore D_6=C_2×D_3 where D_3 is the dihedral group with 6 elements, i.e., the group of symmetries of an equilateral triangle.

There are thus two ways to produce the character table, either inducing from C_2×C_2 and using the orthogonality relations or simply by finding the character tables for D_3 and C_2 and taking their group direct sum.

C_G1x^3xx^2yyx
chi_0111111
chi_11111-1-1
chi_21-1-111-1
chi_31-1-11-11
chi_42-21-100
chi_522-1-100

See also

Dihedral Group

This entry contributed by Declan Davis

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References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 248, 1985.

Cite this as:

Davis, Declan. "Dihedral Group D_6." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/DihedralGroupD6.html

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