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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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If you know it, use inflation/restriction to a 2-Sylow subgroup $S \simeq \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$ of $A_4$, you will find that $H^2(S,\mathbb{Z}/2\mathbb{Z}) \simeq H^2(A_4,\mathbb{Z}/2\mathbb{Z})$ (and this isomorphism is explicit), and $H^2(S,\mathbb{Z}/2\mathbb{Z})$ has eight elements: one corresponding to the direct product $S \times \mathbb{Z}/2\mathbb{Z}$, three to $\mathbb{Z}/4\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$, three to the dihedral group having eight elements, and one to the quaternion $8$-group $D$ ($D/ \{\pm 1 \} \simeq S$). You are interested in the latter (that's the preimage of $S$). It is easy to write this cocyle explicitly. But at that point it is easier to define the binary tetrahedral group as the Hurwitz quaternions units, pick a section, and compute.

Otherwise just take a big piece of paper, and fill an array with $0$s and $1$s.

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    I am confused: The Sylow 2-subgroup of $A_4$ is not cyclic. And this extension has a Quaternion Sylow 2-group, not cyclic.2011-09-13
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    You are right, thank you, I corrected. It's a bit less easy now. Too bad.2011-09-13
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    @Plop : the "fill an array" solution would help for what I have in mind, but is there an easy way to do it ? It's 144 values after all, and I don't feel like checking the cocycle condition for each one of them...2011-09-13
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    @AlexPof: I can show you a short and easy way, but it'll be a few days. Holt's Handbook of CGT describes the method.2011-09-13
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    @Plop: I believe $H^2(S,\mathbb{Z}/2\mathbb{Z})$ has order bigger than 2!2011-09-14
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    @Steve D You are right (I didn't correct that part when I corrected $S$). Correcting it now.2011-09-14
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    @Jack Schmidt : I managed to get a copy of Holt's book, but couldn't find the related section. If you can just indicate the page, I'll have a look at it...2011-09-23
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    @AlexPof: p307-310. I'm writing up an example in pdf. Probably have time to finish it this weekend.2011-09-23