#%Format: latex
#%This file can be pasted directly to GAP
\Read("E6.txt");
#\table
#=(E6+A5+2A2)+A4.1\\\2&\same3 &E8-3333&(1,0)&\eqref{eq.(E6+A5+2A2)+A4.1}%
 \group1l(5,6,3);#%
 #% This returns 7680 vs. 60 for $\Gamma$
 \qsize(g.1^5);#%
 #% Perturbations of torus type
 \group1l(1,6,3);#%
 \simplify(g.1/g.2);#%
 \group1l(5,3,3);#%
 \simplify(g.1/g.2);#%
 #% Perturbations not of torus type
 \size1l(5,2,3);#%
 \size1l(5,6,1);#\cr%
#E6+A13&&E8-9111=\-1&(0,1)&\tref{arg.E6.loop}%
 \size1l(\0,\0,1);#\cr%
#(E6+A11)+A2\\\1&&E8-9111=2&(1,0)&\tref{lem.3.black}%
 #\cr%
#E6+A10+A3&\same4 &E8-9111=3&(1,0)&\tref{arg.E6.loop}%
 \Size(DS(AA(c1l, [11, 4]), loop, A((x*y)^3*x^2, 1)));#\cr%
#E6+A7+A4+A2&\same5 &E8-6321=1&(1,0)&\tref{arg.E6.loop}%
 \size1l(8,5,3);#\cr%
#=(E6+A8+A2)+A3\\\1&&E8-6321=2&(1,0)&\tref{arg.E6.loop.tt}%
 \group1l(4,9,3);#%
 \simplify(g.1/g.2);#\cr%
#E6+A5+2A4&\same6 &E8-6321=3&(1,0)&\tref{arg.E6.loop}%
 \size1l(5,5);#\cr%
#E6+A8+A4+A1&\same7 &E8-6321=4&(0,1)&\tref{arg.E6.loop}%
 \size1l(\0,\0,\0,1);#\cr%
#E6+A10+A2+A1&\same8 &E8-6321=5&(0,1)&\tref{arg.E6.loop}%
 \size1l(\0,\0,1);#\cr%
#=(E6+A8+A2)+A2+A1\\\1&\same9 &E8-6321=6&(1,0)&\tref{arg.E6.loop.tt}%
 \group1l(3,9,\0,3);#%
 \simplify(g.1/g.2);#\cr%
#E6+A7+A6&&E8-5511=\-1&(0,1)&\tref{arg.E6.loop}%
 \size1l(\0,\0,\0,1);#\cr%
#E6+A9+A4&\same10 &E8-5511=\-2&(0,1)&\tref{arg.E6.loop}%
 \size1l(\0,\0,1);#\cr%
#E6+A7+A4+A2&\same5 &E8-5511=3&(1,0)&\tref{arg.E6.loop}%
 \size1l(3,8);#\cr%
#E6+A6+A4+A3&&E8-5511=4&(1,0)&\tref{arg.E6.loop}%
 \size1l(7,4);#\cr%
#E6+A10+A3&\same4 &E8-8211=1&(1,0)&\tref{arg.E6.loop}%
 \size1l(\0,\0,\0,1);#\cr%
#E6+A12+A1&&E8-8211=\-2&(0,1)&\tref{arg.E6.loop}%
 \size1l(\0,\0,1);#\cr%
#E6+A10+A2+A1&\same8 &E8-8211=3&(1,0)&\tref{arg.E6.loop}%
 \size1l(3,11);#\cr%
#E6+D13&&E8-411=\-1&(1,0)&\tref{arg.E6.loop}%
 \size1l(\0,\0,1);#\cr%
#=E6+D5+A8.1&\same11 &E8-411=\-1&(0,1)&\tref{arg.E6.loop}%
 \size1l(\0,\0,1);#\cr%
#=E6+D5+A8.2&\same11 &E8-411=\-1&(1,0)&\tref{arg.E6+D5+A8}%
 \Size(DS(AA(c1l, [9,0,1]), A((y*x^2)^2, 1)));#\cr%
#E6+D11+A2&&E8-411=2&(1,0)&\tref{arg.E6.loop}%
 \size1l(\0,\0,\0,1);#\cr%
#E6+D7+A6&&E8-411=2&(1,0)&\tref{arg.E6.loop}%
 \size1l(\0,\0,\0,1);#\cr%
#=E6+D5+A6+A2.1&\same12 &E8-411=2&(1,0)&\tref{arg.E6.loop}%
 \size1l(3,7);#\cr%
#=E6+D5+A6+A2.2&\same12 &E8-411=2&(1,0)&\tref{arg.E6+D5+A8}%
 \Size(DS(AA(c1l, [3,7,0,1])));#\cr%
#\midrule
#=E6+A9+A4&\same10 &[fig.E6.snail]s9111&(1,0)&\tref{arg.E6.snail}%&l=10
 #\cr%
#E6+A6+A4+A2+A1.1&\same13 &[fig.E6.snail]s6321&(1,0)&\tref{arg.E6.snail}%&l=7
 #\cr%
#E6+A5+2A4&\same6 &[fig.E6.snail]s5511&(1,0)&\tref{arg.E6.snail}%&l=6
 #\cr%
#E6+A8+A4+A1&\same7 &[fig.E6.snail]s8211&(1,0)&\tref{arg.E6.snail}%&l=9
 #\cr%
#=E6+D9+A4&&[fig.E6.snail]s411&(1,0)&\tref{arg.E6.snail}%
 #\cr%
#=E6+D5+2A4&&[fig.E6.snail]s411&(1,0)&\tref{arg.E6.snail}%&l=5
 #\cr%
#\midrule
#=E6+A5+4A2\\\2&&[fig.hex.ir]h336336&(1,0)&\tref{arg.E6.hex}%
 \group0h(3,6,3,3,\0,3);#%
 \qsize(g.1^2);#%> 72
 \StructureDescription(qgroup(g.1^2));#%> (C3 x A4) : C2
 #% Perturbation of weight eight:
 \group0h(3,3,3,3,\0,3);#%
 \qsize(g.1^2);#%> 18
 \StructureDescription(qgroup(g.1^2));#%> (C3 x C3) : C2
 #% This one is considered as a perturbation of (2E6+2A2)+A3
 \group0h(3,2,3,3,\0,3);#%
 #% Other perturbations of torus type
 \group0h(1,6,3,3,\0,3);#%
 \simplify(g.1/g.2);#\cr%
#(E6+A8+A2)+A2+A1\\\1&\same9 &[fig.hex.ir]h993392&(0,1)&\tref{arg.E6.hex}%
 \group0h(2,9,\0,3,3);#%
 \simplify(g.1/g.2);#\cr%
#=(E6+A5+2A2)+A4.2\\\2&\same3 &[fig.hex.ir]h665335&(1,0)&\tref{arg.E6.hex}%
 \group0h(6,5,3,3);#%
 #% This returns 7680 vs. 60 for $\Gamma$
 \qsize(g.1^5);#%
 #% Perturbations of torus type:
 \group0h(3,5,3,3);#%
 \simplify(g.1/g.2);#%
 \group0h(6,1,3,3);#%
 \simplify(g.1/g.2);#\cr%
#=E6+A6+A4+A2+A1.2&\same13 &[fig.hex.ir]h773525&(0,1)&\tref{arg.E6.hex}%
 \Size(group0h(7,3,5,2));#\cr%
#\endtable
#\endinput


## Hexagon with a double loop
r2l := AA([y*x*y*x^2, (y*x)^2*(y*x^2)^2], [5, 1]);;
Size(DS(r2l));
# Returns the same size, hence the extra relation is redundant:
Size(DS(r2l, A(x, 4)));
StructureDescription(h);
# Abelianization splits:
Order(g.1);
Order(g.2);
Order(g.3);
# The product is not direct:
c := Centralizer(g, h);
AbelianInvariants(c);
Size(c);
Size(Intersection(c, h));
# For other curves and perturbations:
Size(DS(r2l, A(x, 2)));
Size(DS(r2l, D(x, 5)));
