According to ATLAS of Group Representations, the alternating group $A_6$ is a group of order 360 which has presentation $$ \langle a,b \mid a^2 = b^4 = (ab)^5 = (ab^2)^5 = 1 \rangle. $$ If we draw its Cayley graph using these two generators, the 4-order element $b$ will induce regular quadrilaterals in the graph. Further if we shrink each quadrilaterals to a vertex, it is not hard to see from the relations that the resulting graph consists of regular pentagons with 6 meeting at each vertex. I've found the graph very similar to 120-cell. However they are different because this graph is expected to have 90 = 360/4 vertices while the 120-cell has 600 ones.
So my questions are:
- Does this graph have a name?
- Is there any picture on the internet that visualizes it?
Edit: it should be 6, instead of 4, pentagons meeting at a vertex