Let $G \subseteq S_8$ be the subgroup generated by some 4-cycles. If we number the elements $1,2,\dots, 8$, the 4-cycles are
- $(1234),(5678),(1485),(2376),(1265),(4378)$
I am not sure if I have the list write... I am thinking of the corners of a cube and rotating them. Let me just write the vertices as elements of $\mathbb{Z}_2^3$. What is the order of this group?
These generators are twists of the cube:
- $\boxed{000 \to 001 \to 011 \to 010}$ $\boxed{100 \to 101 \to 111 \to 110}$
- cyclic permutations of these two by $abc \mapsto bca$
What are the relations among these generators to give a presentation? In the case of the dihedral group we get some relations like:
$$ D_{2n}= \langle r,s \mid r^{n}=s^{2}=1, rs=sr^{-1} \rangle $$
I am looking something similar for my $G$ generated by 4-cycles. Or can this group be an action on a vector space? Right now the twist only works on half the elements.
EDIT The comments suggest $\mathbf{G = S_8}$. Can someone find me a way to get a 2-cycle $(12)$ from these 6 elements?