Regular character (${\chi }_{\mathrm{reg}}(g)=|G|{\delta }_{g,e}$)

The character of the regular representation of a finite group is $|G|$ at the identity and zero elsewhere. Left multiplication permutes its group-indexed basis, and a nonidentity element fixes none of those basis vectors. Thus each complex irreducible representation occurs with representation multiplicity equal to its dimension, by the character inner product.