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 Q_{8}. 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 x^{2}= a^{n}=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 C_{2}. (Neither of these short exact sequences splits.)
Each of the two normal subgroups C_{2n} and C_{2} lead to representations of Dic_{n} from those of C_{2} and D_{n} respectively. The one arising from the non-trivial representation of C_{2} is denoted A_{1}. (Since C_{2} is a subgroup of C_{2n} this coincides with the A_{1} 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 C_{2} (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}.
Dic_{2}
Dic_{2} is isomorphic to Q_{8}. For example, put \(x\mapsto i\) and \(a\mapsto j\), so \(xa\mapsto k\).
Dic_{2} | 1 | x^{2} | a | x | xa | notes |
# | 1 | 1 | 2 | 2 | 2 | |Dic_{2}|=8 |
A_{0} | 1 | 1 | 1 | 1 | 1 | trivial rep |
A_{1} | 1 | 1 | -1 | 1 | -1 | |
A_{2} | 1 | 1 | 1 | -1 | -1 | |
A_{3} | 1 | 1 | -1 | -1 | 1 | |
H | 4 | -4 | 0 | 0 | 0 | quaternionic type |
- H is the representation arising from the identification of Dic_{2} with a subgroup of the unit quaternions (as described above, in fact with Q_{8}). If we identify the quaternions with C^{2} one finds that H is the underlying real representation of the complex 2-dimensional irreducible representation, with character 2,-2,0,0,0.
Dic_{3}
Dic_{3} | 1 | x^{2} | a | a^{2} | x | xa | notes |
# | 1 | 1 | 2 | 2 | 3 | 3 | |Dic_{3}|=12 |
A_{0} | 1 | 1 | 1 | 1 | 1 | 1 | trivial rep |
A_{1} | 1 | 1 | 1 | 1 | -1 | -1 | |
B | 2 | -2 | -2 | 2 | 0 | 0 | complex type |
E | 2 | 2 | -1 | -1 | 0 | 0 | from D_{3} |
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
- For odd values of n, the real representations do not distinguish between {x} and {xa}.
Dic_{4}
Dic_{4} | 1 | x^{2} | a | a^{2} | a^{3} | x | xa | notes |
# | 1 | 1 | 2 | 2 | 2 | 4 | 4 | |Dic_{4}|=16 |
A_{0} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial rep |
A_{1} | 1 | 1 | 1 | 1 | 1 | -1 | -1 | |
A_{2} | 1 | 1 | -1 | 1 | -1 | -1 | 1 | |
A_{3} | 1 | 1 | -1 | 1 | -1 | 1 | -1 | = A_{1}⊗A_{2} |
E | 2 | 2 | 0 | -2 | 0 | 0 | 0 | from D_{4} |
H_{1} | 4 | -4 | 2√2 | 0 | -2√2 | 0 | 0 | quaternionic |
H_{2} | 4 | -4 | -2√2 | 0 | 2√2 | 0 | 0 | quaternionic |
Dic_{5}
Dic_{5} | 1 | x^{2} | a | a^{2} | a^{3} | a^{4} | x | xa | notes |
# | 1 | 1 | 2 | 2 | 2 | 2 | 5 | 5 | |Dic_{5}|=20 |
A_{0} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial rep |
A_{1} | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | |
B | 2 | -2 | -2 | 2 | -2 | 2 | 0 | 0 | complex type |
E_{1} | 2 | 2 | \(\gamma\) | \(\bar\gamma\) | \(\bar\gamma\) | \(\gamma\) | 0 | 0 | from D_{5} |
E_{2} | 2 | 2 | \(\bar\gamma\) | \(\gamma\) | \(\gamma\) | \(\bar\gamma\) | 0 | 0 | from D_{5} |
H_{1} | 4 | -4 | -2\(\bar\gamma\) | 2\(\gamma\) | -2\(\gamma\) | 2\(\bar\gamma\) | 0 | 0 | quaternionic |
H_{2} | 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)
- \(\bar\gamma = 2\cos(4\pi/5) = -\frac12(\sqrt5+1)\)
Dic_{6}
Dic_{6} | 1 | x^{2} | a | a^{2} | a^{3} | a^{4} | a^{5} | x | xa | notes |
# | 1 | 1 | 2 | 2 | 2 | 2 | 2 | 6 | 6 | |Dic_{6}|=24 |
A_{0} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | trivial rep |
A_{1} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -1 | -1 | |
A_{2} | 1 | 1 | -1 | 1 | -1 | 1 | -1 | 1 | -1 | |
A_{3} | 1 | 1 | -1 | 1 | -1 | 1 | -1 | -1 | 1 | |
E_{1} | 2 | 2 | 1 | -1 | -2 | -1 | 1 | 0 | 0 | from D_{6} |
E_{2} | 2 | 2 | -1 | -1 | 2 | -1 | -1 | 0 | 0 | from D_{6} |
H_{1} | 4 | -4 | 2√3 | 2 | 0 | -2 | -2√3 | 0 | 0 | quaternionic |
H_{2} | 4 | -4 | -2√3 | 2 | 0 | -2 | 2√3 | 0 | 0 | quaternionic |
H_{3} | 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 A_{0} ... A_{3};
- n odd: two are real (A_{0} and A_{1}), and two are complex, say B_{1} and B_{2}=B_{1}^{*}, and so the real 2-dimensional rep B = B_{1} + B_{1}^{*} 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).