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# This is the 2 generator Engel-4 group of exponent 5 from Tom Litt's
# thesis. Note that there are free reductions in, eg, [a,ab,ab,ab,ab].
Gr: a,b;
Rel:
(a)^5, (b)^5, (ab)^5, (aab)^5, (abb)^5, (aabb)^5, (Abb)^5, (aB)^5,
(aaB)^5, (aaBB)^5, (aabaabb)^5, (aabaaB)^5, (aabaB)^5, (aabbaBB)^5,
(aabbAAB)^5, (aabAAb)^5, (aabAAbb)^5, (aabAB)^5, (aaBaaBB)^5,
(abaab)^5, (abaaB)^5, (ababb)^5, (abaB)^5, (abbaaBB)^5, (abbaBB)^5,
(abbAbb)^5, (abbAB)^5, (abbABB)^5, (abAb)^5, (abAbb)^5, (abAB)^5,
(AbaaB)^5, (Ababb)^5, (AbbaaBB)^5, (AbbaB)^5, (AbbAAbb)^5, (AbAbb)^5,
(AAbaabb)^5, (AAbbaaB)^5,
[a,b,b,b,b], [A,b,b,b,b], [a,B,B,B,B], [A,B,B,B,B],
[b,a,a,a,a], [B,a,a,a,a], [b,A,A,A,A], [B,A,A,A,A],
[a,ab,ab,ab,ab], [a,AB,AB,AB,AB],
[b,ab,ab,ab,ab], [b,AB,AB,AB,AB]
;
Gen: a,a^b;
Wo:10M;
Mess:25000;
Hard;
# On ozone, yields: INDEX = 15625 (a=15625 r=781 h=1163639 n=1163639;
# l=1510 c=149.99; m=1132447 t=1163638)
Pure C;
# INDEX = 15625 (a=15625 r=1 h=2185271 n=2185271;
# l=2189 c=260.94; m=2005874 t=2185270)
Sims:3;
# INDEX = 15625 (a=15625 r=2908 h=1824088 n=1824088;
# l=5 c=215.02; m=1790527 t=1824087)
Wo:100M;
Mess:100000;
Pure R;
# INDEX = 15625 (a=15625 r=19711329 h=19711329 n=19711329;
# l=34 c=89.46; m=18857618 t=19711328)
# So, some variant of hard seems best ...
Hard; No:0;
# INDEX = 15625 (a=15625 r=611 h=899214 n=899214;
# l=1222 c=119.47; m=867767 t=899213)
# Try all 5th powers up to length 3 (incl.)
Hard; No:8;
# INDEX = 15625 (a=15625 r=425 h=625261 n=625261;
# l=850 c=84.01; m=598641 t=625260)
# Now let's try some greedy relator deletions ...
Group Relators:
(a)^5, (b)^5,
(ab)^5, (aB)^5,
(aab)^5, (abb)^5, (Abb)^5, (aaB)^5,
(aabb)^5, (aaBB)^5, (abaB)^5, (abAb)^5, (abAB)^5,
(aabaB)^5, (aabAB)^5, (abaab)^5, (abaaB)^5, (ababb)^5, (abbAB)^5,
(abAbb)^5, (AbaaB)^5, (Ababb)^5, (AbbaB)^5, (AbAbb)^5,
# (aabaaB)^5, (aabAAb)^5, (abbaBB)^5, (abbAbb)^5, (abbABB)^5,
AbabABabAbaBABaBAbabABaBAbaBABab,
abAbaBAbabABaBABabAbaBABabABaBAb,
ABaBAbaBABabAbabABaBAbabABabAbaB,
aBABabABaBAbabAbaBABabAbaBAbabAB,
BabaBAbaBabABAbABabaBAbABabABAba,
baBabABabaBAbABAbaBabABAbaBAbABa,
BAbABabABAbaBabaBAbABabaBAbaBabA,
bABAbaBAbABabaBabABAbaBabABabaBA,
baabABabbaBAABBAbaabABBAbaBAABab,
ABBAbaBAABabbaabABBAbaabABabbaBA,
abbaBAbaabABBAABabbaBAABabABBAba,
BAABabABBAbaabbaBAABabbaBAbaabAB;
# (aabaabb)^5, (aabbaBB)^5, (aabbAAB)^5, (aabAAbb)^5, (aaBaaBB)^5,
# (abbaaBB)^5, (AbbaaBB)^5, (AbbAAbb)^5, (AAbaabb)^5, (AAbbaaB)^5;
Wo:4M;
Mess:50000;
Hard; No:8; DMod:0;
% INDEX = 15625 (a=15625 r=520354 h=520354 n=520354;
% l=848 c=53.56; m=498065 t=520353)
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