Consider $D_8 \times Q_8$, where $D_8$ is the dihedral group of order 8; $Q_8$ the quaternions. Let $F$ be a field of characteristic not equal to 2.

What is the smallest dimension possible for a representation of $D_8 \times Q_8$ which is faithful over $F$?

NB: It doesn't need to irreducible. I have so far worked out that in fact it cannot be irreducible and faithful, using Schur's Lemma.


Short answer: it depends. Either 4 or 8, depending on whether the value for $Q_8$ is 2 or 4, respectively. If the field is finite, the answer is always 4 (and 2 for $Q_8$).

Long answer:

The representations of $D_8 \times Q_8$ of the form $X \otimes Y$ for $X$ a representation of $D_8$ and $Y$ a representation of $Q_8$. Such a representation is faithful if and only if both of $X$ and $Y$ are faithful.

Over a field of characteristic not 2, all of the representations are completely reducible. If a representation contains a sub-representation with multiplicity higher than 1, then we can omit all but one copy without changing whether or not the representation is faithful. In particular, we need only consider sums of distinct irreducible representations.

$D_8$ is very well behaved: all of its absolutely irreducible representations are defined over the prime field. A sum of irreducible representations is faithful iff it involves the 2-dimensional representation. Hence, the degrees of its faithful representations are $2+k$ for $k \geq 0$.

$Q_8$ is slightly more complicated: Over each field it has a unique minimal faithful representation, but the dimension of that representation is a little hard to calculate: it is either $2$ or $4$. Thus the degrees of its faithful representations are $2_F + l$ for $l \geq 0$ and $2_F \in \{2,4\}$ depends only on $F$.

The minimum value of the product $(2+k)(2_F+l)$ is $22_F$ which is either $4$ or $8$ depending on $F$.

Here are a few values of $2_F$: If $F$ is finite, then $2_F = 2$. If $F=\mathbb{Q}$ then $2_F = 4$. If $F=\mathbb{Q}[i]$ then $2_F = 2$.

The number $2_F / 2$ is called the Schur index (of that two-dimensional absolutely irreducible representation of $Q_8$) over $F$.


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