Dicyclic groups

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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 Dn 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 C2n 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 C2n and C2 lead to representations of Dicn from those of C2 and Dn respectively. The one arising from the non-trivial representation of C2 is denoted A1. (Since C2 is a subgroup of C2n this coincides with the A1 representation of Dn.)

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

{e}, {x2} (each with 1 element),
{a, a-1}, {a2, a-2}, . . ., {an-1, an+1} (each with 2 elements), and
{x, xa2, . . . xa2n-2}, {xa, xa3, . . . xa2n-1} (each with n-1 elements).

A simple argument shows that over C, Dicn 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: ak and a2n-k are conjugate, as are xak and xa2n-k, and x and x-1.

Dic2

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

Dic2 1 x2 a x xa notes
# 1 1 2 2 2 |Dic2|=8
A0 1 1 1 1 1 trivial rep
A1 1 1 -1 1 -1
A2 1 1 1 -1 -1
A3 1 1 -1 -1 1
H 4 -4 0 0 0 quaternionic type
  • 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

Dic3 1 x2 a a2 x xa notes
# 1 1 2 2 3 3 |Dic3|=12
A0 1 1 1 1 1 1 trivial rep
A1 1 1 1 1 -1 -1
B 2 -2 -2 2 0 0 complex type
E 2 2 -1 -1 0 0 from D3
H 4 -4 -2 2 0 0 quaternionic

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
1, -1, -1, 1, i, -i, and its conjugate.
  • For odd values of n, the real representations do not distinguish between {x} and {xa}.

Dic4

Dic4 1 x2 a a2 a3 x xa notes
# 1 1 2 2 2 4 4 |Dic4|=16
A0 1 1 1 1 1 1 1 trivial rep
A1 1 1 1 1 1 -1 -1
A2 1 1 -1 1 -1 -1 1
A3 1 1 -1 1 -1 1 -1 = A1⊗A2
E 2 2 0 -2 0 0 0 from D4
H1 4 -4 2√2 0 -2√2 0 0 quaternionic
H2 4 -4 -2√2 0 2√2 0 0 quaternionic

Dic5

Dic5 1 x2 a a2 a3 a4 x xa notes
# 1 1 2 2 2 2 5 5 |Dic5|=20
A0 1 1 1 1 1 1 1 1 trivial rep
A1 1 1 1 1 1 1 -1 -1
B 2 -2 -2 2 -2 2 0 0 complex type
E1 2 2 \(\gamma\) \(\bar\gamma\) \(\bar\gamma\) \(\gamma\) 0 0 from D5
E2 2 2 \(\bar\gamma\) \(\gamma\) \(\gamma\) \(\bar\gamma\) 0 0 from D5
H1 4 -4 -2\(\bar\gamma\) 2\(\gamma\) -2\(\gamma\) 2\(\bar\gamma\) 0 0 quaternionic
H2 4 -4 -2\(\gamma\) 2\(\bar\gamma\) -2\(\gamma\) 2\(\gamma\) 0 0 quaternionic

Notes

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

Dic6

Dic6 1 x2 a a2 a3 a4 a5 x xa notes
# 1 1 2 2 2 2 2 6 6 |Dic6|=24
A0 1 1 1 1 1 1 1 1 1 trivial rep
A1 1 1 1 1 1 1 1 -1 -1
A2 1 1 -1 1 -1 1 -1 1 -1
A3 1 1 -1 1 -1 1 -1 -1 1
E1 2 2 1 -1 -2 -1 1 0 0 from D6
E2 2 2 -1 -1 2 -1 -1 0 0 from D6
H1 4 -4 2√3 2 0 -2 -2√3 0 0 quaternionic
H2 4 -4 -2√3 2 0 -2 2√3 0 0 quaternionic
H3 4 -4 0 -4 0 4 0 0 0 quaternionic

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 Ei) arising from the homomorphism Dicn → Dn, and there are quaternionic representations (here denoted Hi) from Dicn → SU(2).