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I was thinking of defining a semidirect product in this way: $$7^{1+2}_+ \rtimes 3^{1+2}_+$$

Recalling that $7^{1+2}_+ \simeq (C_7 \times C_7 )\rtimes C_7$, the idea was to define $$\phi: 3^{1+2}_+ \to 3^{1+2}_+ / Z(3^{1+2}_+) \simeq C_3 \times C_3 \to \operatorname{Aut}(C_7 \times C_7)$$

where each $C_3$ acts on the respective $C_7$ separately. Then, we would extend the same action to $(C_7 \times C_7) \rtimes C_7$ just by "ignoring" the third $C_7$ (this is the part of which I am unsure).

However, I am confused by the structure of these extraspecial groups and I really don't understand what I should check in this case to see if such a map is well-defined.

So, my question is: do the above definitions give a well-defined semidirect product between the two extraspecial groups? And, if they don't, is there any way of defining such a thing (excluding, of course, the direct product)?

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Presentations are useful for checking whether assignements of generators extend to group automorphisms.

Your group $(C_7 \times C_7) \rtimes C_7$ has a presentation $$\langle a,b,c \mid a^7=b^7=c^7=1,[a,c]=[b,c]=1,[b,c]=a \rangle,$$ where $a$, $b$ and $c$ can be taken as the generators of the three $C_7$ subgroups.

An assignment of generators extends to a homomorphism if and only if it preserves the relations of the presentation. This applies in particular to the assignments $$\alpha:a \mapsto a^{2}, b \mapsto b,c \mapsto c^{2},$$ and $$\beta:a \mapsto a^2, b \mapsto b^{2},c \mapsto c,$$ and so they both extend to homomorphisms, which are clearly automorphisms, and they both have order $3$.

Note also that $$\alpha\beta = \beta \alpha:a \mapsto a^4, b \mapsto b^{2},c \mapsto c^2,$$ so $\langle \alpha,\beta \rangle \cong C_3 \times C_3$, which is what you want.

In fact $\langle \alpha,\beta \rangle$ is a Sylow $3$-subgroup of ${\rm Aut}((C_7 \times C_7) \rtimes C_7)$, so this is up to isomorphism the only such semidirect product with faithful action.

But note that the assignment, which might have been one of the possibilities that you were considering, $$a \mapsto a^2, b \mapsto b,c \mapsto c,$$ does not extend to an automorphism.

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  • $\begingroup$ I hadn't thought about using presentations. Very simple and clear, thank you very much! $\endgroup$ – FifteenPointOne Jun 6 '17 at 17:55

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