# Groups with special character degrees

Let $G$ be a finite group of order $p_1^{a_1}\times p_2^{a_2}\times\cdots \times p_n^{a_n}$. Is there any classification for simple groups such that for each $i$, $p_i^{a_i}$ is an irreducible character degree of $G$?

Any comments or hints is highly appreciated.

• These should be rare, though there are some examples which rely on arithmetical accidents, such as ${\rm PSL}(2,5)$ and ${\rm PSL}(2,7)$. In general Fermat or Mersene primes $p$ such that $\frac{p+1}{2}$ or $\frac{p-1}{2}$ respectively are also primes give more examples. – Geoff Robinson Feb 27 '16 at 11:01
• It may be interesting that if we choose one irreducible character of each of these degrees and take their product, we necessarily get the regular character of G. – Marty Isaacs Mar 20 '16 at 20:23
• It is not hard to prove that the only alternating groups with this property are $A_5$ and $A_6$. Use the fact that for $n>5$, the smallest character degree of $A_n$ is $n-1$ and $|A_n|$ has a prime divisor $p$ in the range $[n/2,n-2]$ (so its multiplicity is 1). – Glasby Jan 4 '17 at 9:48