Is there an average distance between groups of galaxies such as our Local Group? I believe our Local Group is $10$ million light years in diameter and, depending on which article I read, has $30-52$ galaxies.  But what about other galaxy groups? How far would they be?  
I calculated that if our Milky Way was one inch long then the Andromeda galaxy would be $2.2$ inches long and $25$ inches away. Our Local Group would be about $8.3$ feet in diameter. So I am wondering how far apart, on average, are these "galaxy groups".  If I know the distance in light years or millions of light years I can convert that to something I can visualize so I can imagine in my mind what the large scale structure looks like. I have searched but can't seem to find any scale models.
 A: Here is a picture of the nearby galaxies in the Karachentsev catalog:

This is a view from the Local Void direction; most of the galaxy clusters lie along a rough plane (the Local Sheet). The Local Group is in the middle, and the closest groups are about 4 Mpc away. 
Generally there is a bit of confusion between galaxy groups, galaxy clusters, and larger structures like superclusters. Galaxy groups are collections that are gravitationally bound, typically 1-2 Mpc away from each other. Clusters are larger (2-10 Mpc) and contain several groups, while superclusters are collections of groups and clusters (150-200 Mpc across). At the larger scales defining where one thing begins and ends is somewhat arbitrary, especially since they are no longer gravitationally bound over cosmological time. Filaments and sheets of galaxies make the edges and distances even fuzzier. 
So while there will always be some average depending on what definitions one chose, the distances will be fairly dependent on the definition and where one is in the universe.
See also this paper and visualisations for some more mapping. 
(One Mpc = 3.26 million lightyears)
A: You can get a rough estimate by looking at the size and contents of the supercluster we are a part of, Laniakea. Wikipedia lists its major axis as being $159\operatorname{Mpc}$ and it has $300$–$500$ clusters inside of it. Roughly, the volume per cluster will be
\begin{align}
    V_C &= \frac{V_{\mathrm{Laniakea}}}{N_{\mathrm{Laniakea}}} \\
      &\approx \frac{\frac{4}{3} \pi (79.5\operatorname{Mpc})^3}{300\ \mathrm{to}\ 500} \\
      & = (4.2\ \mathrm{to}\ 7.0) \times 10^3 \operatorname{Mpc}^3.
\end{align}
This estimate of the volume per cluster is probably an underestimate, since our census of the clusters in Laniakea is likely incomplete. Regardless, the average distance between clusters is estimated as twice the cluster radius
\begin{align}
    D & \approx 2 R_C \\
      & = 2 \sqrt[3]{\frac{3}{4\pi} V_C} \\
      & = (20 \ \mathrm{to}\ 24) \operatorname{Mpc}.
\end{align}
For reference, the radius of the milky way is $15$ to $28\operatorname{kpc}$ and the distance to the Andromeda galaxy is $780\operatorname{kpc}$.
Note that this is the typical size of clusters, like the Virgo cluster, and not the size of "local groups", like the Milky Way/Andromeda system. For the size of local groups, I would imagine that doubling the distance to andromeda would give a reasonable typical size estimate, $1.5\operatorname{Mpc}$. Given that clusters usually contain hundreds to thousands of galaxies, having them be a factor of $10$ bigger than local groups sounds reasonable.
