/*
This text file is readable by Magma, and it
can be downloaded using your browser. This Magma file 
is self-contained.

There are many comments describing the data. All these
comments are in the form of string variables beginning with the
word "comment_". Note that if you read two of these data files
with Magma, the information from the first file will be overwritten 
by the second one, since the names of the variables are the same.
*/


/**************************************************************/
/**************************************************************/
// This is a data file readable by magma.

group_order := 16;
hall_number := 11;
G := FreeGroup(Ilog2(group_order));
field<a> := GF(4);
one_var_ring := PolynomialRing(Integers());
hilbert_ring<T> := FieldOfFractions(one_var_ring);
master_polynomial_ring<z,y,x,w,v,u,t,s,r,q,p,n,
m,k,j,i,h,g,f,e,d,c,b> := PolynomialRing(field, 23);
string_variables := ["z","y","x",
"w","v","u","t","s","r","q","p","n",
"m","k","j","i","h","g","f","e","d","c","b"];
AssignNames(~master_polynomial_ring, string_variables);

isomorphism_type := "";
magma_number := 6;
pcgroup_relations := [
    G.1^2 = G.2, 
    G.2^2 = G.4, 
    G.3^G.1 = G.3 * G.4
 ];
G := quo< G | pcgroup_relations >;

centre_type :=  [ 4 ]
;
orders_lower_central_series :=  [ 16, 2, 1 ]
;
orders_upper_central_series :=  [ 1, 4, 16 ]
;
order_of_frattini_subgroup := 4;
exponent := 8;
orders_of_max_elem_abelian_subs := [ 4 ];
orders_of_centralizers_of_max_elem_abelian_subs := [ 8 ];
orders_of_normalizers_of_max_elem_abelian_subs := [ group_order ];
variables := [ 
    z,
    y,
    x,
    w
 ];
degrees := [  1, 1, 3, 4  ];
relations := [ 
    z^2,
    z*y^2,
    z*x,
    x^2
 ];
hilbert_series := 
1/(T^4 - 2*T^3 + 2*T^2 - 2*T + 1);
krull_dimension := 2;
depth := 1;
longest_regular_sequence := [ 
    w
 ];
homogeneous_set_of_parameters := [ 
    w,
    y^2
 ];
param_degrees :=  [ 4, 2 ];



comment_HYPERCOHOMOLOGY_SPECTRAL_SEQUENCE_DATA :=
"[ list, boolean ]
Each element of the list is of the form
<vector(as string), several lists of polynomials>
and describes a row of the spectral sequence. The boolean is true
if and only if Poincare duality is satisfied.";

hypercohomology_spectral_sequence := [* 
< "(1)", [],
[
    z
],
[
    z*y
]>,
< "(0)", [1],
[
    y,
    z
],
[
    z*y
],
[
    x
],
[
    y*x
]>, true *];



comment_MAXIMAL_SUBGROUPS_DATA :=
"This is a list, in which each entry represents
a maximal subgroup. Each item on the list has:
1: maximal subgroup number,
2: generators of the maximal subgroup (in the larger group),
3: Hall-Senior number of its isomorphism class,
4: list of images (in the cohomology ring) under restriction,
5: list of generators of the ideal of the kernel of restriction,
6: list of generators of its cohomology ring,
7: list of images under the transfer map.";

maximal_subgroups_data := [
< 1,  [ G.4, G.1 * G.3 * G.4, G.2 ]
,
3,
[ 
    z,
    z,
    z*y,
    y^2
 ],
[ 
    z + y
 ],
[ 
    y
 ],
[ 
    z*y
 ] >,
< 2,  [ G.3, G.4, G.2 ]
,
2,
[ 
    0,
    y,
    z*y^2,
    y^2*x + x^2
 ],
[ 
    z
 ],
[ 
    z,
    x,
    z*x
 ],
[ 
    z,
    z*y + y^2,
    x
 ] >,
< 3,  [ G.1, G.4, G.2 ]
,
3,
[ 
    z,
    0,
    z*y,
    y^2
 ],
[ 
    y
 ],
[ 
    y
 ],
[ 
    z*y
 ] > ];



comment_MAXIMAL_ELEMENTARY_ABELIAN_SUBGROUPS_DATA :=
"Two lists are computed.\n
The first list contains the maximal elementary abelian subgroups
which are also maximal subgroups; in this case, their index numbers are
identical (that is, the n-th maximal elementary abelian is the n-th maximal).\n
The second list contains items describing
the maximal elementary abelian subgroups:
1: maximal elementary abelian subgroup number,
2: generators of the subgroup (in the larger group),
3: order,
4: list of images (in the cohomology ring) under restriction,
5: list of generators of the ideal of the kernel of restriction,
6: list of generators of its cohomology ring,
7: list of images under the transfer map.

We also compute information about the nilradical. This is the list:
1: string (description of the nilradical),
2: generators of the nilradical,
3: nilpotency degree,
4: boolean (is the nilradical zero?).";

maximal_elementary_abelian_and_maximal := [  ];
maximal_elementary_abelian_subgroups_data := [
< 1,  [ G.4, G.3 * G.4 ]
,
4,
[ 
    0,
    y,
    0,
    z^4 + z^2*y^2
 ],
[ 
    z,
    x
 ],
[ 
    z,
    z^2,
    z^3
 ],
[ 
    z,
    0,
    x
 ] >
 ];

nilradical := [*
"The kernel of the restriction to E is also the nilradical of P", [ ],
2, false *];



comment_ESSENTIAL_COHOMOLOGY_DATA := 
"This is a tuple < > containing the following entries:
1: generators of the image of transfers from all maximal subgrops,
2: boolean (is the essential cohomology zero?),
3: generators of the essential cohomology,
4: nilpotency degree,
5: dimension of the annihilator,
6: generators of the subring,
7: generators of the free module";

essential_cohomology_data := < [ 
    z,
    y^2,
    x
 ],
false, [ 
    z*y
 ],
2, 1, [ 
    w
 ],
[ 
    z*y
 ] >;



comment_DEPTH_ESSENTIAL_COHOMOLOGY_DATA := 
"We compute two lists.
The first list contains generators of the centralizers.\n
The second list consists of the following items:
1: generators of the depth essential cohomology,
2: dimension of the annihilator,
3: generators of the subring,
4: generators of a free module,
5: statement about DI,
6: generators of DI. ";

list_of_depth_centralizers := [
[  G.2 * G.3 * G.4, G.3  ] ];
depth_essential_cohomology_data := [*
[ 
    z
 ],
1, [ 
    w
 ],
[ 
    z,
    z*y
 ],
"The ideal DI is the image of the transfer from the unique centralizer",
[ ] *];



comment_ASSOCIATED_PRIME_IDEALS :=
"One list is computed. Each entry contains the following:
1: associated prime number,
2: maximal elementary abelian subgroup number (a number; 0 if N/A),
3: generators (empty if item 3 is non-zero),
4: [ element ] (whose annihilator is the associated prime); this is   given as a list with one element (or empty if unavailable)."; 

associated_primes_data := [*
< 1, 1, [ ], [ 
    x
 ] >,
< 2, 0, [  G.4  ],
[ 
    z*y
 ] > *];



comment_INFLATIONS_FROM_MAXIMAL_QUOTIENT_GROUPS_DATA :=
"This is one list. Each entry has the following information:
1: maximal quotient group number,
2: generator of the kernel of the quotient map,
3: Hall-Senior number of its isomorphism class,
4: list of images (in the cohomology ring) under inflation,
5: generators of the kernel of the inflation.";

inflations_from_maximal_quotient_groups_data := [
< 1,  G.4 , 2, [ 
    z,
    y,
    z*y
 ],
[ 
    z*y + x,
    y*x
 ] > ];



comment_ACTION_OF_AUTOMORPHISMS_DATA := 
"We compute the order of the group of automorphisms,
the number of generators of the automorphism group,
and a list containing information about the generators
of the group of automorphisms.
Each entry of the list has the following information:
1: automorphism number,
2: order in the outer automorphism group,
3: list of images of the group generators under this automorphism,
4: list of images of the generators on cohomology under the
     map induced by the automorphism.";

order_of_outer_automorphism_group := 4;
number_of_generators_of_outer_automorphism_group := 2 
;
automorphisms_data := [
< 1, 2,
[  G.1 * G.2, G.2 * G.4, G.3, G.4  ],
[ 
    z,
    y,
    x,
    w
 ] >,
< 2, 2,
[  G.1 * G.3, G.2 * G.4, G.3, G.4  ],
[ 
    z,
    z + y,
    x,
    w
 ] > ];


super_groups := 
[ 13, 13, 13, 13, 17, 17, 17, 31, 32, 32, 44, 45, 47, 47, 48, 48 ]
;



super_quots := 
[ 13, 13, 19, 19, 20, 21 ]
;