Theory

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Further details

• S₃ S₄

Dicyclic groups

On this page

• Introduction
• Dic₂ = Q₈ - the quaternion group
• Dic₃
• Dic₄
• Dic₅
• Dic₆

Introduction

The dicyclic group is defined by $$\mathrm{Dic}_n = \left<a,\,x \mid a^{2n}=e,\; x^2=a^n,\; xax^{-1} = a^{-1}\right>.$$ It follows from this that \(x\) has order 4 and \(a\) order \(2n\), and \(\mathrm{Dic}_n\) has order \(4n\). For n=2, \(\mathrm{Dic}_2\) is the quaternion group Q₈. Indeed, \(\mathrm{Dic}_n\) can be viewed as a subgroup of the unit quaternions, by putting \(x = j\) and \(a=\mathrm{e}^{i\pi/n}\).

The centre of \(\mathrm{Dic}_n\) is of order 2, generated by \(x^2 = a^n\), corresponding to -I in SU(2)), leading to $$0 \longrightarrow C_2 \longrightarrow \mathrm{Dic}_n \longrightarrow D_n \longrightarrow 0,$$ (where D_n is the dihedral group of order 2n, generated by a and x with x²= aⁿ=1). Presented in this way \(\mathrm{Dic}_n\) is seen to be the binary dihedral group.

The other normal subgroup (containing the centre) is the cyclic subgroup C_2n generated by a, with $$0 \longrightarrow C_{2n} \longrightarrow \mathrm{Dic}_n \longrightarrow C_2 \longrightarrow 0.$$ The element x projects to the nontrivial element of C₂. (Neither of these short exact sequences splits.)

Each of the two normal subgroups C_2n and C₂ lead to representations of Dic_n from those of C₂ and D_n respectively. The one arising from the non-trivial representation of C₂ is denoted A₁. (Since C₂ is a subgroup of C_2n this coincides with the A₁ representation of D_n.)

These are all (isomorphic to) subgroups of the infinite group \(\mathrm{Dic}_\infty\), which is the subgroup of SU(2) generated by j and all \(\mathrm{e}^{i\theta}\). This has centre C₂ (generated by -I) and satisfies $$0 \longrightarrow S^1 \longrightarrow \mathrm{Dic}_\infty \longrightarrow C_2 \longrightarrow 0.$$

Conjugacy Classes

In Dic_n there are n+3 conjugacy classes, the first two forming the centre of the group. They are:

A simple argument shows that over C, Dic_n has 4 1-dimensional representations. If n is even these are all real, while if n is odd two of these are complex conjugates, and their sum is then a real irreducible representation of dimension 2, of complex type (denoted B in the tables below).

Character tables

Note: a^k and a^2n-k are conjugate, as are xa^k and xa^2n-k, and x and x^-1.

Dic₂

Dic₂ is isomorphic to Q₈. For example, put \(x\mapsto i\) and \(a\mapsto j\), so \(xa\mapsto k\).

• H is the representation arising from the identification of Dic₂ with a subgroup of the unit quaternions (as described above, in fact with Q₈). If we identify the quaternions with C² one finds that H is the underlying real representation of the complex 2-dimensional irreducible representation, with character 2,-2,0,0,0.

Dic₃

Notes

• H is the underlying real rep of the complex rep with character 2,-2,-1,1,0,0. And H=B⊗ E.
• B is the sum of two complex irreducibles, with characters

• For odd values of n, the real representations do not distinguish between {x} and {xa}.

Dic₄

Dic₅

Notes

• \(\gamma = 2\cos(2\pi/5) = \frac12(\sqrt5-1)\) (=golden ratio)
• \(\bar\gamma = 2\cos(4\pi/5) = -\frac12(\sqrt5+1)\)

Dic₆

General n

The pattern continues: for all n there are four 1-dimensional irreducible complex representations:

• n even: these are all real, denoted A₀ ... A₃;
• n odd: two are real (A₀ and A₁), and two are complex, say B₁ and B₂=B₁^*, and so the real 2-dimensional rep B = B₁ + B₁^* is of complex type.

There are then representations (denoted E_i) arising from the homomorphism Dic_n → D_n, and there are quaternionic representations (here denoted H_i) from Dic_n → SU(2).