1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
|
# run this file with
# gap -b -q < /dev/null group.gap | perl -pe 's/\\\n//s' | indent -kr
Print("/* ----- data generated by group.gap begins ----- */\n\n");
Print("struct group {\n unsigned long autosize;\n");
Print(" int order, ngens;\n const char *gens;\n};\n");
Print("struct groups {\n int ngroups;\n");
Print(" const struct group *groups;\n};\n\n");
Print("static const struct group groupdata[] = {\n");
offsets := [0];
offset := 0;
for n in [2..26] do
Print(" /* order ", n, " */\n");
for G in AllSmallGroups(n) do
# Construct a representation of the group G as a subgroup
# of a permutation group, and find its generators in that
# group.
# GAP has the 'IsomorphismPermGroup' function, but I don't want
# to use it because it doesn't guarantee that the permutation
# representation of the group forms a Cayley table. For example,
# C_4 could be represented as a subgroup of S_4 in many ways,
# and not all of them work: the group generated by (12) and (34)
# is clearly isomorphic to C_4 but its four elements do not form
# a Cayley table. The group generated by (12)(34) and (13)(24)
# is OK, though.
#
# Hence I construct the permutation representation _as_ the
# Cayley table, and then pick generators of that. This
# guarantees that when we rebuild the full group by BFS in
# group.c, we will end up with the right thing.
ge := Elements(G);
gi := [];
for g in ge do
gr := [];
for h in ge do
k := g*h;
for i in [1..n] do
if k = ge[i] then
Add(gr, i);
fi;
od;
od;
Add(gi, PermList(gr));
od;
# GAP has the 'GeneratorsOfGroup' function, but we don't want to
# use it because it's bad at picking generators - it thinks the
# generators of C_4 are [ (1,2)(3,4), (1,3,2,4) ] and that those
# of C_6 are [ (1,2,3)(4,5,6), (1,4)(2,5)(3,6) ] !
gl := ShallowCopy(Elements(gi));
Sort(gl, function(v,w) return Order(v) > Order(w); end);
gens := [];
for x in gl do
if gens = [] or not (x in gp) then
Add(gens, x);
gp := GroupWithGenerators(gens);
fi;
od;
# Construct the C representation of the group generators.
s := [];
for x in gens do
if Size(s) > 0 then
Add(s, '"');
Add(s, ' ');
Add(s, '"');
fi;
sep := "\\0";
for i in ListPerm(x) do
chars := "ABCDEFGHIJKLMNOPQRSTUVWXYZ";
Add(s, chars[i]);
od;
od;
s := JoinStringsWithSeparator([" {", String(Size(AutomorphismGroup(G))),
"L, ", String(Size(G)),
", ", String(Size(gens)),
", \"", s, "\"},\n"],"");
Print(s);
offset := offset + 1;
od;
Add(offsets, offset);
od;
Print("};\n\nstatic const struct groups groups[] = {\n");
Print(" {0, NULL}, /* trivial case: 0 */\n");
Print(" {0, NULL}, /* trivial case: 1 */\n");
n := 2;
for i in [1..Size(offsets)-1] do
Print(" {", offsets[i+1] - offsets[i], ", groupdata+",
offsets[i], "}, /* ", i+1, " */\n");
od;
Print("};\n\n/* ----- data generated by group.gap ends ----- */\n");
quit;
|
