#%Format: latex
#%This file can be pasted directly to GAP
\Read("D5.txt");
#\table
#\splitting{C_3+C_3\quad(G/[G,G]=\CG4\oplus\CG3)}
#=D5+A7+A5+A2\\\1&&E7-6321=1&(1,0)&\tref{arg.D5.r}\iref{i.D5+A7+A5+A2}%
 \size(6,8,3);#%
 \StructureDescription(h);#%
 #% Perturbations:
 \size(3,8,3);#%
 \size(6,1,3);#\cr%
#D5+A9+A3+A2&&E7-6321=1&(1,0)&\tref{arg.D5.r}%
 \size(10,4,3);#\cr%
#D5+A11+A2+A1&&E7-6321=4&(0,1)&\tref{arg.D5.r}%
 \size(12,\0,1);#\cr%
#D5+A13+A1&&E7-8211=2&(0,1)&\tref{arg.D5.r}%
 \size(14,\0,1);#\cr%
#D5+D14&&E7-411=1&(1,0)&\tref{arg.D5.r}%
 \Size(DS(AA(lbb, [0,0,1]), D(x^2, 10), bb));#\cr%
#=2D5+A9\\\1&&E7-411=1&(0,1)&\tref{arg.D5.r}\iref{i.2D5+A9}%
 \size(10,\0,1);#%
 #% Perturbations:
 \size(5,\0,1);#\cr%
#\midrule\splitting{C_4+C_2\quad(G/[G,G]=\CG2\oplus\CG2)}
#=D5+2A5+A4\\\1&&E7-6321=2&(1,0)&\tref{arg.D5.r}\iref{i.D5+2A5+A4}%
 \size(6,5,6);#%
 \AbelianInvariants(h);#%
 \IdFunc(NumberPerfectGroups(Size(h)));#%
 #% Perturbations:
 \size(3,5,6);#%
 \size(6,5,3);#\cr%
#D5+A9+A4+A1&&E7-6321=\-3&(0,1)&\tref{arg.D5.r}%
 \size(10,5,2);#\cr%
#=D5+A9+2A2+A1\\\1&&E7-6321=5&(1,0)&\tref{arg.D5.r}\iref{i.D5+A9+2A2+A1}%
 \size(10,3,\0,3);#%
 #% Perturbations:
 \size(5,3);#\cr% %% The other one degenerates to E7-9111
#=D5+A11+A3\\\1&&E7-8211=1&(1,0)&\tref{arg.D5.r}\iref{i.D5+A11+A3}% >>
 \size(12,4,\0,1);#%
 \AbelianInvariants(h);#%
 #% Perturbations:
 \size(3,4,\0,1);#%
 \size(12,1,\0,1);#\cr%
#=D5+A11+A2+A1\\\1&&E7-8211=3&(1,0)&\tref{arg.D5.r}\iref{i.D5+A11+A2+A1}%
 \group(12,3,\0,2);#%
 \simplify(g.2/g.3);#%
 #% Perturbations:
 \size(3,3,\0,2);#%
 \size(12,3,\0,1);#\cr%
#D5+D12+A2&&E7-411=2&(1,0)&\tref{arg.D5.r}%
 \size(\0,3,\0,1);#\cr%
#=D5+D7+A7\\\1&&E7-411=2&(1,0)&\tref{arg.D5.r}\iref{i.D5+A11+A3}%
 \size(8,\0,\0,1);#%
 \AbelianInvariants(h);#%
 #% Perturbations:
 \size(1);#\cr%
#=2D5+A7+A2\\\1&&E7-411=2&(1,0)&\tref{arg.2D5+A7+A2}%
 \group(8,3,8);#%
 \AbelianInvariants(h);#%
 #% Perturbations:
 \size(1);#\cr%
#\midrule\splitting{C_5+C_1\quad(G/[G,G]=\CG8)}
#D5+A6+A5+A3&&E7-6321=2&(1,0)&\tref{arg.D5.r}%
 \size(7,4);#\cr%
#D5+A7+A6+A1&&E7-6321=\-3&(0,1)&\tref{arg.D5.r}%
 \size(7,8,2);#\cr%
#=D5+A7+A4+A2+A1\\\1&&E7-6321=5&(1,0)&\tref{arg.D5.r}\iref{i.D5+A7+A4+A2+A1}%
 \size(5,8,\0,\0,3);#%
 \IdFunc(NumberPerfectGroups(Size(h)));#%
 \IdFunc(PerfectGroup(Size(h)));#%
 #% Perturbations:
 \size(5,1);#\cr%  %% The other one degenerates to E7-9111
#D5+A9+A5&&E7-8211=1&(1,0)&\tref{arg.D5.r}%
 \size(6,10,\0,\0,1);#\cr%
#D5+A9+A4+A1&&E7-8211=3&(1,0)&\tref{arg.D5.r}%
 \size(5,10,\0,\0,2);#\cr%
#D5+D10+A4&&E7-411=2&(1,0)&\tref{arg.D5.r}%
 \size(5,\0,\0,\0,1);#\cr%
#D5+D9+A5&&E7-411=2&(1,0)&\tref{arg.D5.r}%
 \size(\0,6,\0,\0,1);#\cr%
#2D5+A5+A4&&E7-411=2&(1,0)&\tref{arg.D5.r}%
 \size(5,6);#\cr%
#\endtable
#\endinput


## Further processing for 2D5+A7+A2 (E7-411=2)
group(8,3,8);
simplify(g.2/g.3);
TzPrintRelators(P);
#I  1. _x3^-2*_x7^-1*_x3^3*_x7*_x3^-1
# ...
# this relation implies that _x3^3 -> h.1^3 is central
h := FpGroupPresentation(P);
hh := FactorGroupFpGroupByRels(h, [h.1^3]);
AbelianInvariants(hh);  # => h.1^3 has infinite order in h/[h,h]
StructureDescription(DerivedSubgroup(hh));  # >> Q8


##===================================================================
## One component perturbations of two component curves
##===================================================================

#D5+A7+A5+2A1&&E7-4422=1 % size(8,6,2)
 size(1,6,2);
 size(8,3,2);  # >> \CG3
 size(8,6,1);
#D5+A7+2A3+A1&&E7-4422=2 % size(8,4,4)
 size(1,4,4);
 size(8,4,1);
# All others degenerate to E7-8211

#D5+D10+A3+A1&&E7-222 % size(\6,4,2)
 size(0,1,2);
 size(0,4,1);
 size(1,4,2);
 size(3,4,2);
 size(5,4,2);
#D5+D8+A5+A1&&E7-222 % size(6,\4,2)
 size(3,0,2);
 size(6,0,1);
 size(3,1,2);
 size(3,3,2);
#D5+D6+A5+A3&&E7-222 % size(6,4,\2)
 size(3,4,0);  # >> \SL(2,\FF5)
 size(6,1,0);
 size(6,4,1);

#D5+D8+D6&&E7-00 % size(\4,\2) 
 size(1,1);
 size(3,1);

##===================================================================
## The other skeletons (\autoref{fig.d5-r})
##===================================================================

# \autoref{fig.d5-r}(a)
#D5+2A7
 inf(x*y*x^2); # (8,0,8,0)
 bg := Union(bg, A((y*x^2)^2, 1));;  # from a monogon
 size2(1,0,8);
 size2(8,0,1);
#D5+A9+A5
 inf(x*y*x^2); # (10,6)
 bg := Union(bg, A((x*y)^3*x^2, 1));; # from a monogon
 size2(5,6);
 size2(10,3);
#D5+A7+A5+A2
 inf((x*y)^2*x^2); # (8,6)
 bg := Union(bg, A(x*y*x^2, 3));; # from the triangle
 size2(1,6);
 size2(8,3);

# \autoref{fig.d5-r}(b)
#D5+A11+A3
 inf(x*y*x^2); # (4,12)
 bg := Union(bg, A(y*x*(y*x^2)^2, 1));;  # from a monogon
 size2(1,12);
 size2(4,3);
#D5+A13+A1
 inf(x*y*x^2); # (14,2)
 bg := Union(bg, A((x*y)^3*x^2, 1));; # from a monogon
 size2(7,2);
 size2(14,1);
#D5+A11+A2+A1
 inf((x*y)^2*x^2); # (12,2)
 bg := Union(bg, A(x*y*x^2, 3));; # from the triangle
 size2(3,2);
 size2(12,1);

# \autoref{fig.d5-r}(c)
 inf(x*y*x^2); # (4,6,2)  (three components)
#D5+D10+A3+A1  # (4,\6,2)
 size2(1,1,2);
 size2(1,3,2);
 size2(1,5,2);
 group2(1,0,1);; AbelianInvariants(g); # >> [0] : still reducible
 size2(4,1,1);
 size2(4,3,1);
 size2(4,5,1);
#D5+D8+A5+A1   # (\4,6,2)
 size2(1,3,2);
 size2(3,3,2);
 group2(1,6,1);; AbelianInvariants(g); # >> [0] : still reducible
 size2(0,3,1);
#D5+D6+A5+A3   # (4,6,\2)
 size2(1,3,0);
 group2(1,6,1);; AbelianInvariants(g); # >> [0] : still reducible
 size2(4,3,1);

# \autoref{fig.d5-r}(d)
# The irreducible perturbations of the other insertion
# degenerate to fig.(h)
#D5+A5+A4+A3+A2
 inf(x*y*x^2); # (6,3,4,5)
 size2(3,3,4,5);
 size2(6,3,1,5);

# \autoref{fig.d5-r}(e)
# The perturbations of the central bigon degenerate to fig.(b)
#D5+A7+A5+2A1
 inf(x*y*x^2); # (8,2,0,6) (three components)
 size2(1,1,0,6);
 size2(1,2,0,3);
 group2(8,1,0,3);; AbelianInvariants(g); # >> [0] : still reducible
#D5+A7+2A3+A1
 inf((x*y)^2*x^2); # (8,2,0,6) (three components)
 size2(1,1,0,6);
 size2(1,2,0,3);
 group2(8,1,0,3);; AbelianInvariants(g); # >> [0,2] : still reducible
#D5+2A5+A3+A1
 inf((x*y)^2*x^2); # (6,6,2) (three components)
 size2(3,3,2);
 group2(3,6,1);; AbelianInvariants(g); # >> [0] : still reducible
 size2(6,3,1);
#D5+A7+2A3+A1
 inf(x*y*x^2); # (4,8,4) (three components)
 size2(1,1,4);
 group2(1,8,1);; AbelianInvariants(g); # >> [0] : still reducible
 size2(4,1,1);

# \autoref{fig.d5-r}(f)
#D5+A9+A3+A2
 inf(x*y*x^2); # (10,3,0,4)
 size2(5,3,0,4);
 group2(10,3,0,1);; AbelianInvariants(g); # >> [0] : still reducible
#D5+A9+A4+A1
 inf(x*y*x^2); # (10,2,0,5)
 size2(5,2,0,5);
 group2(10,1,0,5);; AbelianInvariants(g); # >> [0] : still reducible
#D5+A11+A2+A1
 inf((x*y)^2*x^2); # (12,2,0,3)
 size2(3,2,0,3);
 group2(12,1,0,3);; AbelianInvariants(g); # >> [0,2] : still reducible
#D5+A9+2A2+A1
 inf((x*y)^3*x^2); # (10,2,0,3)
 size2(5,2,0,3);
 group2(10,1,0,3);; AbelianInvariants(g); # >> [0] : still reducible

# \autoref{fig.d5-r}(g)
#D5+A6+A5+A3
 inf(x*y*x^2); # (7,6,0,4)
 size2(7,3,0,4);
 size2(7,6,0,1);
#D5+A7+A6+A1
 inf(x*y*x^2); # (8,7,2)
 size2(1,7,2);
 size2(8,7,1);
#D5+A9+A4+A1
 inf((x*y)^2*x^2); # (10,5,2)
 size2(5,5,2);
 size2(10,5,1);
#D5+A7+A4+A2+A1
 inf((x*y)^3*x^2); # (8,5,2)
 size2(1,5,2);
 size2(8,5,1);
