Transitive groups of small degree

Alexander Hulpke

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In work starting with my PhD thesis, I have classified the transitive groups up to degree 30. The list of groups will be made available in GAP.

This list contains corrected, final counts. (Preliminary results disagreed in degrees 24,27 and 28.)

Numbers of Groups

Degree 2 3 4 5 6 7 8 9 10 11

prim. 1 2 2 5 4 7 7 11 9 8
trans. 1 2 5 5 16 7 50 34 45 8
fpf. 2 2 7 8 37 40 200 258 1039 1501
Total 2 4 11 19 56 96 296 554 1593 3094
                     
Degree 12 13 14 15 16 17 18 19 20 21

prim. 6 9 4 6 22 10 4 8 4 9
trans. 301 9 63 104 1954 10 983 8 1117 164
                     
Degree 22 23 24 25 26 27 28 29 30 31

prim. 4 7 5 28 7 15 14 8 4 12
trans. 59 7 25000 211 96 2392 1854 8 5712 12

All Numbers refer to classes up to Sn-conjugacy:

 

prim.:
Number of primitive groups
trans.:
Number of transitive groups (including primitive)
fpf.:
Number of fix point free groups (i.e. groups that are not subgroups of smaller symmetric groups.
Total:
Total(n-1)+fpf(n)

Alexander Hulpke (hulpke@math.colostate.edu)