The repetition code encodes $\vert \psi \rangle = \alpha \vert 0 \rangle + \beta \vert 1 \rangle \rightarrow \vert \psi \rangle = \alpha \vert 000 \rangle + \beta \vert 111 \rangle$ using the following circuit:
where $$t_0: \vert \psi \rangle = (\alpha \vert 0 \rangle + \beta \vert 1 \rangle) \otimes \vert 0 \rangle \otimes \vert 0 \rangle = \alpha \vert 000 \rangle + \beta \vert 100 \rangle$$ $$t_1: \vert \psi \rangle = \alpha \vert 000 \rangle + \beta \vert 110 \rangle$$ $$t_2: \vert \psi \rangle = \alpha \vert 000 \rangle + \beta \vert 111 \rangle$$.
I am wondering what is the operator used from $t_0$ to $t_1$? I understand that an XOR was applied there to get the desired output but what is the explicit operator (gate-wise or matrix-wise) used?
Furthermore, I tried $U = \mathbb{I} \otimes \mathbb{X} \otimes \mathbb{I}$ and applied it to $\vert \psi \rangle$ which resulted in $$\vert \psi \rangle = \alpha \vert 010 \rangle + \beta \vert 110 \rangle$$