# Minimal generating set for Dih4 x Z2

I'm trying to find the minimal generating set for a square prism under reflection. This group is $$\text{Dih}_4\times Z_2$$.

Geometrically, the set $$\{a,b,c\}$$ where $$a$$ is a rotation of $$90^\circ$$ about the axis through the square faces, $$b$$ is a rotation of $$180^\circ$$ about an axis through two rectangular faces, and $$c$$ is a reflection about some plane, would generate the group. After all, $$\{a,b\}$$ generates $$\text{Dih}_4$$ and $$\{c\}$$ generates $$Z_2$$. However, I don't know how to prove that there does not exist two elements $$\{x,y\}$$ which generate the whole group.

Is this even the case? In any case, how can it be proven or disproven?

• Can you show that the group you have has a quotient that is isomorphic to $Z_2\times Z_2\times Z_2$? – Arturo Magidin Jun 5 at 1:48
• I think so; is it sufficient to show that $Z_2\times Z_2$ is a quotient of $\text{Dih}_4$? – Ovinus Real Jun 5 at 2:16
• Yes, since then you get that $Z_2\times Z_2\times Z_2$ is a quotient of $D_4\times Z_2$. So, given that, suppose you had a $2$-element generating set for $D_4\times Z_2$. Would the images of the elements generate any quotient? If so, can $Z_2\times Z_2\times Z_2$ be generated by $2$ elements? – Arturo Magidin Jun 5 at 2:43
• I see, so the logic is if some set $S\subseteq D_4\times Z_2=G$ can generate $G$, and there is a normal subgroup $H$ such that $|G/H|=Z_2\times Z_2\times Z_2$, then the cosets of $H$ with respect to $S$ also generate $G/H$, so since the minimal generating set of $Z_2\times Z_2\times Z_2$ clearly has cardinality $3$, we have $|\text{cosets}|\geq 3\Longrightarrow |S|\geq 3$? – Ovinus Real Jun 5 at 3:10
• It’s not that the number of cosets is at least $3$ (that assertion is unclear). It’s that any generating set for $G$ maps to a generating set for $G/N$ for any normal subgroup $N$ of $G$. So if you know that there is an such that $G/N$ needs at least $k$ generators, then $G$ needs at least $k$ generators as well (though it may need more). – Arturo Magidin Jun 5 at 3:23