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Since the binary tetrahedral group is an extension of $\mathbb{Z}_2$ by $A_4$, the group product between elements $(h_1,g_1) \cdot (h_2,g_2), h_1,h_2 \in \mathbb{Z}_2, g_1,g_2 \in A_4$ can be expressed as : $$(h_1 h_2 \chi(g_1,g_2),g_1\cdot g_2)$$ with $\chi : A_4 \times A_4 \to \mathbb{Z}_2$ a 2-cocycle.

How can I find a detailed expression/values for such a cocycle ? Thanks for your help...

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