Transitive Group Data

All data below was computed with GAP. Data is given for transitive subgroups of Sn, for various small n. Suppose F is a field, K is a degree n separable extension of F, and Kg is a Galois closure of K/F. Then Gal(Kg/F) is one of the groups "nTj" below. The T-numbering is standard for transitive subgroups of Sn and is implemented GAP.

Data presented here is sometimes described in group terms, and sometimes in terms of the fields F, K, and Kg with the Galois correspondence understood. For each group, we give

• Its T-number
• Group names from Conway, John H.; Hulpke, Alexander; McKay, John. On transitive permutation groups. LMS J. Comput. Math. 1 (1998), 1--8
• The order of the group, which equals |Aut(Kg/F)| = [Kg:F]
• The parity: 1 if the group is a subgroup of An and -1 otherwise
• The order of the centralizer of G in Sn. This is also equal to the order of Aut(K/F)
• Subfield information for K when n is not prime. We consider each proper intermediate field between F and K with degree d over F, up to isomorphism, and give the Galois group of its normal closure as a dTj for some j.
• Other representations as transitive permutation groups. The entries are listed as dTj where d is the degree and j is the corresponding T number. These are listed with repetition corresponding to subfields of Kg up to F-isomorphism. So for example, 4T3 lists itself as an "other representation" because the extension Kg/F has two isomorphism classes of quartic subfields with the same Galois closure. The group is the dihedral group of order 8, so it also is listed as 8T4. Under 8T4, there will be correspondingly two entries of 4T3 listed as other representations.
• Resolvents gives proper subfields E of Kg which are non-trivial Galois extensions over F, sorted by the degree [E : F]. These are listed with multiplicity up to F-isomorphism. For each such Galois field, we give it in terms of its smallest degree transitive permutation representation (and if needed, smallest T-number among isomorphic groups).

Degree 23

Last Update: May 31, 2012