Assessing Group Similarities and Dissimilarities Post PCA
What does the distance between groups mean for your problem? Answering that problem would help you pick a distance metric.  

Assuming you pick e.g. Euclidean distance, one simple way to find inter-group distance would be to first calculate the group centroid, i.e. average location of each group by averaging the locations of each group together.  You can do this by just averaging all the 5x1 vectors belonging to a group, and repeat for each group.  Then calculate the Euclidean distance between the resulting 6 centroids, giving you a 6x6 distance matrix.