An irreducible nonmonomial representation of degree eight in a semiabelian group of order 2592

Abstract

We construct a semiabelian group \(G\) of order \(2592\) with an irreducible complex representation of degree eight which is not induced from a linear character of any subgroup. Write \(M=C_2^3\rtimes A_4\) as a signed permutation group and choose \(K\leq M\) with \(K\cong Q_8\rtimes C_3\) and \([M:K]=4\). The group \(G\) is the semidirect product of the augmentation module of \(\mathbb F_3[M/K]\) by \(M\). A coordinate character and the irreducible quaternionic representation of \(K\) give the degree-eight representation. Any monomial realization would produce a subgroup of index two in \(K\), which does not exist. This disproves the assertion in Kourovka Problem 21.68 that every finite semiabelian group is monomial.