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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
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Its T-number
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Group names from Conway, John H.; Hulpke, Alexander; McKay, John.
On transitive permutation groups. LMS J. Comput. Math. 1 (1998), 1--8
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The order of the group, which equals
|Aut(Kg/F)| = [Kg:F]
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The parity: 1 if the group is a subgroup of An
and -1 otherwise
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The order of the centralizer of G
in Sn. This is also equal to the order of
Aut(K/F)
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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.
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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.
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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).
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