Sorry to reply almost three months later. I opened #24469 for that,
with a link to this discussion.
https://trac.sagemath.org/ticket/24469
Sun 2017-10-15 16:25:11 UTC, John Cremona:
>
> Extracting information about a Galois group is more painful than it
> should be. After
>
> sage: K.<z> = CyclotomicField(5)
> sage: G = K.galois_group(type='pari')
> sage: G
> Galois group PARI group [4, -1, 1, "C(4) = 4"] of degree 4 of the
> Cyclotomic Field of order 5 and degree 4
>
> we have
>
> sage: type(G)
> <class 'sage.rings.number_field.galois_group.GaloisGroup_v1'>
>
> (other types are returned if other options for the galois_group()
> method are chosen). There is not a lot you can do with this G except
> get its order (G.order()) without going deeper:
>
> sage: GG=G.group()
> sage: type(GG)
> <class 'sage.groups.pari_group.PariGroup_with_category'>
> sage: GG
> PARI group [4, -1, 1, "C(4) = 4"] of degree 4
>
> This type has "forgotten" that it is a Galois group but has many more
> methods; sadly most not implemented. At least one might want to
> extract the 4 elements of the underlying list which are the order (4)
> which in this example happens to also be the degree (4), meaning that
> GG is a subgroup of S_4 (degree=4) of order 4. The second entry -1 is
> the sign (-1 means odd, i.e. not a subgroup of A_4), the third is the
> "T-number" which identifies this group in some classification of
> transitive groups.
>
> As far as I know the only way to get the sign and T-number is to
> retrieve the underlying PARI list via GG.__pari__() (which until
> recently was GG._pari_() with single underscores). I would like to
> implement
>
> GG.sign() # returns GG.__pari__()[1]
> GG.t_number() # returns GG.__pari__()[2]
>
> and perhaps more. I have been looking in the PARI/gp documentation on
> Galois groups and what it says about this 4-tuple is
>
> "The output is a 4-component vector [n,s,k,name] with the following
> meaning: n is the cardinality of the group, s is its signature (s = 1
> if the group is a subgroup of the alternating group A_d, s = -1
> otherwise) and name is a character string containing name of the
> transitive group according to the GAP 4 transitive groups library by
> Alexander Hulpke.
>
> k is more arbitrary and the choice made up to version 2.2.3 of PARI is
> rather unfortunate: for d > 7, k is the numbering of the group among
> all transitive subgroups of S_d, as given in "The transitive groups of
> degree up to eleven", G. Butler and J. McKay, Communications in
> Algebra, vol. 11, 1983, pp. 863--911 (group k is denoted T_k there).
> And for d ≤ 7, it was ad hoc, so as to ensure that a given triple
> would denote a unique group. Specifically, for polynomials of degree d
> ≤ 7, the groups are coded as follows, using standard notations (etc)"
>
> Despite the ad hoc nature of this parameter k I still think we should
> allow users to get at it more easily.
>
> John
>
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