Comparing groups of 16S rRNA sequences - how to?

I have several groups of 16S rRNA sequences associated with taxonomic groups (let’s call them A, B and C). The sequences within each group have a common ancestor and are on average more closely related to sequences within the same group. The groups are also related, but less.

Ideally I’d like to have a table like this (where percentages represent the average similarity):

           Group A     Group B     Group C
Group A      99%                  
Group B      97%         98%              
Group C      94%         95%         99%

Is there software that can calculate these percentages for me? I’m thinking that I could set up a many-against-many BLAST search and then try to use pandas in python in order to parse the resulting table…But if something like this already exists, I’d rather not reinvent the wheel. Thank you!

makeblastdb -in 16S_sequences.fasta -dbtype nucl -out my_16S_sequences_db

blastn -db my_16S_sequences_db -query 16S_sequences.fasta -outfmt 6 
       -out allvsall_16S.tsv -num_threads "$(nproc)"