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We define the superclasses for a classical finite unipotent group $U$ of type$B_{n}(q)$, $C_{n}(q)$, or $D_{n}(q)$, and show that, together with thesupercharacters defined in a previous paper, they form a supercharacter theory.In particular, we prove that the supercharacters take a constant value on eachsuperclass, and evaluate this value. As a consequence, we obtain afactorization of any superclass as a product of elementary superclasses. Inaddition, we also define the space of superclass functions, and prove that itis spanned by the supercharacters. As as consequence, we (re)obtain thedecomposition of the regular character as an orthogonal linear combination ofsupercharacters. Finally, we define the supercharacter table of $U$, and provevarious orthogonality relations for supercharacters (similar to the well-knownorthogonality relations for irreducible characters).

A supercharacter theory for the Sylow p-subgroups of the finite symplectic and orthogonal groups