Degree of irreducible representation need not divide exponent

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Statement

We can have a finite group and an irreducible linear representation of the group over an algebraically closed field of characteristic zero (or more generally over a splitting field), such that the degree of the irreducible representation does not divide the exponent of the group.

This is a non-constraint on the Degrees of irreducible representations (?) of a finite group.

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Proof

Example of big extraspecial groups

Consider an extraspecial group of order p^7 for any prime p. The exponent of this group is either p or p^2. On the other hand, it admits a faithful irreducible representation of degree p^3.

For odd primes p, we can take an example of an extraspecial group of order p^5 and exponent p, which admits a faithful irreducible representation of degree p^2.

Example of symmetric group of degree six

Further information: symmetric group:S6, linear representation theory of symmetric group:S6

The symmetric group of degree six has an irreducible representation of degree 16 over the rational numbers, arising from the self-conjugate partition 6 = 3 + 2 + 1. On the other hand, the exponent of the group is the lcm of the numbers from 1 to 6, which is 60, and is not a multiple of 16.