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

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.