Maximum characters in a deterministic Turing machine

Assume we have a deterministic Turing machine $ M = (q_s, q_a, q_r, \Sigma, \Gamma, \delta, Q, b)$ where $ q_s,q_a,q_r$ are the (unique) starting state, accept state and reject state respectively, $ Q$ the set of non-final states, $ \Sigma$ the input alphabet, $ \Gamma$ the tape alphabet, $ \delta$ the transition function and $ b \in \Gamma$ the blank symbol.

How many characters can fit in $ \Gamma$ , as a function of $ |\Sigma|, |Q|$ such that for each $ c \in \Gamma$ , $ \delta$ will be defined by it for some state and character?