Abstract: The integer sequences with first term $ 1$ comprise a group $ \mathcal{G}$ under convolution, namely, the Appell group, and the lower triangular infinite integer matrices with all diagonal entries $ 1$ comprise a group $ \mathbb{G}$ under matrix multiplication. If $ A\in \mathcal{G}$ and $ M\in \mathbb{G},$ then $ MA\in \mathcal{G}.$ The groups $ %% \mathcal{G}$ and $ \mathbb{G}$ and various subgroups are discussed. These include the group $ \mathbb{G}^{(1)}$ of matrices whose columns are identical except for initial zeros, and also the group $ \mathbb{G}^{(2)}$ of matrices in which the odd-numbered columns are identical except for initial zeros and the same is true for even-numbered columns. Conditions are determined for the product of two matrices in $ \mathbb{G}^{(m)}$ to be in $ \mathbb{G}%% ^{(1)}. $ Conditions are also determined for two matrices in $ \mathbb{G}%% ^{(2)}$ to commute.