't Hooft anomaly matching condition and chiral symmetry breaking without bilinear condensate

We explore 4-dimensional SU(N) gauge theory with a Weyl fermion in an irreducible self-conjugate representation. This theory, in general, has a discrete chiral symmetry. We use 't Hooft anomaly matching condition of the center symmetry and the chiral symmetry, and find constraints on the spontaneous chiral symmetry breaking in the confining phase. The domain-walls connecting different vacua are discussed from the point of view of the 't Hooft anomaly. We consider the SU(6) gauge theory with a Weyl fermion in the rank 3 anti-symmetric representation as an example. It is argued that this theory is likely to be in the confining phase. The chiral symmetry ${\mathbb{Z}}_6$ should be spontaneously broken to ${\mathbb{Z}}_2$ under the assumption of the confinement, although there cannot be any fermion bilinear condensate in this theory.