Logging to /Users/s/tmp/pari-10.19
GPRC Done.

        GP/PARI CALCULATOR Version 2.12.1 (development 25476-78c5f8d2c)
          i386 running darwin (x86-64/GMP-6.2.0 kernel) 64-bit version
     compiled: Jun 21 2020, Apple clang version 11.0.0 (clang-1100.0.33.17)
                            threading engine: single
                 (readline v8.0 enabled, extended help enabled)

                     Copyright (C) 2000-2020 The PARI Group

PARI/GP is free software, covered by the GNU General Public License, and comes 
WITHOUT ANY WARRANTY WHATSOEVER.

Type ? for help, \q to quit.
Type ?17 for how to get moral (and possibly technical) support.

parisizemax = 900001792, primelimit = 1000000
(16:37) gp > ?polgalois
polgalois(T): Galois group of the polynomial T (see manual for group coding). Return [n, s, k, 
name] where n is the group order, s the signature, k the index and name is the GAP4 name of 
the transitive group.

(16:41) gp > ??polgalois
polgalois(T):

   Galois group of the non-constant polynomial T\inQ[X]. In the present version 2.12.1, T must
be irreducible and the degree d of T must be less than or equal to 7.   If the galdata package
has  been installed,  degrees 8,  9,  10 and 11 are also implemented.   By definition,  if K =
Q[x]/(T),   this  computes  the action of the Galois group of the Galois closure of K on the d
distinct roots of T, up to conjugacy (corresponding to different root orderings).

   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

   In degree 1: S_1 = [1,1,1].

   In degree 2: S_2 = [2,-1,1].

   In degree 3: A_3 = C_3 = [3,1,1], S_3 = [6,-1,1].

   In degree 4: C_4 = [4,-1,1], V_4 = [4,1,1], D_4 = [8,-1,1], A_4 = [12,1,1], S_4 =
[24,-1,1].

/*-- (type RETURN to continue) --*/
   In  degree  5: C_5 = [5,1,1],  D_5 = [10,1,1],  M_{20} = [20,-1,1],  A_5 = [60,1,1],  S_5 =
[120,-1,1].

   In  degree 6: C_6 = [6,-1,1],  S_3 = [6,-1,2],  D_6 = [12,-1,1],  A_4 = [12,1,1],  G_{18} =
[18,-1,1],  S_4^ -= [24,-1,1],  A_4 x C_2 = [24,-1,2], S_4^ += [24,1,1], G_{36}^ -= [36,-1,1],
G_{36}^ += [36,1,1], S_4 x C_2 = [48,-1,1], A_5 = PSL_2(5) = [60,1,1], G_{72} = [72,-1,1], S_5
= PGL_2(5) = [120,-1,1], A_6 = [360,1,1], S_6 = [720,-1,1].

   In  degree  7:  C_7  =  [7,1,1],  D_7 = [14,-1,1],  M_{21} = [21,1,1],  M_{42} = [42,-1,1],
PSL_2(7) = PSL_3(2) = [168,1,1], A_7 = [2520,1,1], S_7 = [5040,-1,1].

   This  is  deprecated  and  obsolete,  but for reasons of backward compatibility,  we cannot
change  this  behavior  yet.    So  you  can  use the default new_galois_format to switch to a
consistent  naming  scheme,   namely k is always the standard numbering of the group among all
transitive subgroups of S_n.  If this default is in effect, the above groups will be coded as:

   In degree 1: S_1 = [1,1,1].

   In degree 2: S_2 = [2,-1,1].

   In degree 3: A_3 = C_3 = [3,1,1], S_3 = [6,-1,2].

   In degree 4: C_4 = [4,-1,1], V_4 = [4,1,2], D_4 = [8,-1,3], A_4 = [12,1,4], S_4 =
[24,-1,5].

   In  degree  5: C_5 = [5,1,1],  D_5 = [10,1,2],  M_{20} = [20,-1,3],  A_5 = [60,1,4],  S_5 =
[120,-1,5].

   In  degree 6: C_6 = [6,-1,1],  S_3 = [6,-1,2],  D_6 = [12,-1,3],  A_4 = [12,1,4],  G_{18} =
/*-- (type RETURN to continue) --*/
[18,-1,5],  A_4 x C_2 = [24,-1,6],  S_4^ += [24,1,7], S_4^ -= [24,-1,8], G_{36}^ -= [36,-1,9],
G_{36}^ += [36,1,10], S_4 x C_2 = [48,-1,11], A_5 = PSL_2(5) = [60,1,12], G_{72} = [72,-1,13],
S_5 = PGL_2(5) = [120,-1,14], A_6 = [360,1,15], S_6 = [720,-1,16].

   In  degree  7:  C_7  =  [7,1,1],  D_7 = [14,-1,2],  M_{21} = [21,1,3],  M_{42} = [42,-1,4],
PSL_2(7) = PSL_3(2) = [168,1,5], A_7 = [2520,1,6], S_7 = [5040,-1,7].

Warning.  The  method  used  is  that of resolvent polynomials and is sensitive to the current
precision.  The precision is updated internally but, in very rare cases, a wrong result may be
returned if the initial precision was not sufficient.

   The library syntax is GEN polgalois(GEN T, long prec).  To enable the new format in library
mode, set the global variable new_galois_format to 1.