Exponent of the direct sum of finite groups, specifically, $\sum^t_i S_{N_i} \wr D_{m_i}$

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I have one general and one specific questions.

  1. What is the expression for the exponent of the direct sum of finite groups?
  2. What is the exponent of $\sum^t_i S_{N_i} \wr D_{m_i}$?

Here, $i, N_i, m_i$ are positive integers, $S_{N_i}$ is the symmetric group over $N_i$ symbols, $D_{m_i}$ is the dihedral group of order $2 m_i$.

My effort: I guess the exponent should be the lcm of the exponent of each $S_{N_i} \wr D_{m_i}$.

Am I correct?