Characters and dimensions I was looking at one of the problems in the book, "Algebra" by Artin which states: Find the dimension of the irreducible representations of the octahedral group, dihedral groups $D_5$ and $D_6$. From, what I remember, the dimension can be determined by the character table. But what can be the character table of the octahedral group? It has no particular relations. Also, for $D_5$ and $D_6$ is there another way to compute the dimension other than the character table?

Okay, so this will only work on groups of small order.

$D_5$ has only four conjugacy classes: They are comprised, respectively, of 1) identity 2) two rotations 3) two other rotations 4) five reflections

This means $D_5$ has exactly four irreducible characters. Let their degrees be $d_1=1$, $d_2$, $d_3$, $d_4$. Assume $d_{i}\le d_{i+1}$.

Then $10 = |D_5|= 1^2 + d_2^2 + d_3^3 +d_4^2$.

The only possibility is $d_2=1$, $d_3=2$, $d_4=2$.

Let me add: If you want to apply this technique to $D_6$ and the octahedral group, note that $D_6$ has has order $12$ and $6$ conjugacy classes, and the octahedral group has order $24$ and $5$ conjugacy classes.