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Find all complex irreducible representations of the group $\mathbb{H}_8 \times C_2^2$

I am learning representation theory. We studied some useful theorems such as Maschke's theorem and Schur's theorem which say that a representation of a group over $\mathbb{C}$ can be decomposed as a sum of representation of degree $1$. All of this is interesting, but it does not help in finding all irreducible representation of a group.

That's why for fun I tried to find all irreductible representations of the group $ G = \mathbb{H}_8 \times C_2^2$. The problem is that I don't know at all how to do this. Is it even possible?

I know that there is a whole theory about finding irreductible representation of symmetric groups through Specht module. So I guess that in order to find all irreducible representations of this group I need to find an embedding of $G$ in $S_n$ for some $n$ and then use all the theory about the symmetric to get the irreducible representation of $G$. An other idea would be to find the number of conjugacy class of $G$ since it will give me the number of representations of $G$, but once again I don't see how it helps. Knowing the number of representations doesn't seem to really help in findind them.

I would love to know what are the general strategies to find the irreducible representation of a group that is not abelian and not symmetric, just as $G$.

Thank you!

N.B.: $\mathbb{H}_8$ is the Quaternions.

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    $\begingroup$ What is the group $\mathbb{H}_8$? Is that the Quaternions? $\endgroup$ – Sam Hughes Feb 5 '19 at 16:18
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    $\begingroup$ @SamHughes Yes this is the Quaternion group $\endgroup$ – dghkgfzyukz Feb 5 '19 at 16:32
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    $\begingroup$ Are you familiar with tensor products? If you have a direct product of two groups and the irreducible representations of each of them, then the representations of the direct product can be obtained by tensoring together irreducible representations of each group. If you're familiar with group character theory, it maybe be easier to do this by obtaining the character table of $\mathbb{H}_8\times C^2_2$ from the character tables of $\mathbb{H}_8$ and $C_2^2$. $\endgroup$ – Sam Hughes Feb 5 '19 at 16:35
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    $\begingroup$ @SamHughes Thank you for your answer. I am not familar with tensor product but we briefly see this yesterday. So if I get a representation $p$ of $\mathbb{H}_8$ and $r$ of $C_2^2$ then $r \otimes c$ is a representation of $\mathbb{H}_8 \times C_2^2$ ? $\endgroup$ – dghkgfzyukz Feb 5 '19 at 16:38
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    $\begingroup$ I'll write all of this an answer with some extended remarks, but yes start with the character tables, so you at least know what you are looking for (dimension wise). Then think about reconstructing the representations. $\endgroup$ – Sam Hughes Feb 5 '19 at 16:47
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If you have a direct product of two groups and the irreducible representations of each of them, then the representations of the direct product can be obtained by tensoring together irreducible representations of each group (a representation $p$ of $Q_8$ and $r$ of $C^2_2$ then $r\otimes p$ is a representation of $H_8\times C_2^2$).

It is certainly worth calculating the character tables first, then this discussion should be helpful (see the answer) for reconstructing the faithful irreducible representations of $Q_8$ (there are 2, the other representations are all linear one-dimensional).

From there it should not be too much effort to calculate the tensor products. If you get stuck calculating the character table of $Q_8$ there are plenty of examples online (e.g. my masters project).

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  • $\begingroup$ Thank you very much ! Sorry but I made a typo in my comment it should $r \otimes p$ and not $r \otimes c$. That's a nice pdf you wrote there! $\endgroup$ – dghkgfzyukz Feb 5 '19 at 17:02
  • $\begingroup$ No problem! I should have checked that more carefully! $\endgroup$ – Sam Hughes Feb 5 '19 at 17:03
  • $\begingroup$ I don't know anything about representation theory but I am wondering : does having the irreductible representations of $Q_8 \times C_2^2$ helps understanding geometrically this group ? For example I know that we can think of $S_3$ as all linear applications that leave stable an equilateral triangle in the plane. Is it possible to have a similar result for this group ? (getting a geometric intuition about the group through the irredcutible representations of it) ? Thank you. $\endgroup$ – Thinking Feb 6 '19 at 20:27

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