Trying to get a better understanding of dicyclic groups, I wonder what the structure of symmetries of a napkin-ring is.
Suppose you paint a pattern on the ring that has $n$-fold rotational symmetry and also up-down and left-righ symmetry, like $n$ periods of a sine wave that runs around the ring. Then the rotational symmetry is generated by an element $a$ of order $n$, where $n=26$ in the following depictions using the uppercase letters $\text{A⋅⋅⋅Z}$: $\def\pp#1{{\style{display: inline-block; transform:scale(+1,+1)}{\text{#1}}}}$ $\def\np#1{{\style{display: inline-block; transform:scale(-1,+1)}{\text{#1}}}}$ $\def\pn#1{{\style{display: inline-block; transform:scale(+1,-1)}{\text{#1}}}}$ $\def\nn#1{{\style{display: inline-block; transform:scale(-1,-1)}{\text{#1}}}}$
Then $a^k$ generates the 26 pure rotations, and $\{a^k\}\simeq \Bbb Z/n\Bbb Z$:
$$\pp{ABC···XYZ}\\ \pp{BCD···YZA}\\ \vdots\\ \pp{ZAB···WXY}\tag{00}$$
Adding an element $x$ of order 2 turns the group into the dihedral group $D_{2n}$, where $x$ in the depiction is turning the ring upside down: $a^kx$ generates
$$\nn{ABC···XYZ}\\ \nn{BCD···YZA}\\ \vdots\\ \nn{ZAB···WXY}\tag{10}$$
We can add yet another element $y$ of order 2, like the action of a vertical mirror. In two dimensions and with a flat $n$-gon, we'd have $y=a^mx$ for some $m$, but in 3D with the ring this is not the case, and we get $2n$ more symmetries: $a^ky$ are the mirror images of $a^k$:
$$\np{ABC···XYZ}\\ \np{BCD···YZA}\\ \vdots\\ \np{ZAB···WXY}\tag{01}$$
and $a^kxy$ are the mirror images of $a^kx$ resp. upside-down reflections of $a^k$:
$$\pn{ABC···XYZ}\\ \pn{BCD···YZA}\\ \vdots\\ \pn{ZAB···WXY}\tag{11}$$
So what's the group structure of these symmetries $S$? As far as I understand, we have
$$ \langle a,x\rangle \simeq \langle a,y\rangle \simeq \langle a,xy\rangle \simeq \langle a,yx\rangle \simeq D_{2n} \tag 2 $$
and that $S \not\simeq D_{4n}$ and $S \not\simeq Q_{4n}$ for $n$ not too small, and where $Q_{4n}$ denotes the dicyclic group of order $4n$. That $S\not\simeq Q_{4n}$ I infer from this post that states that dicyclic groups are "not easy to visualize". And for $Q_{4n}$ we'd need an element $z\neq a^k$ of order 4, which we don't have.