The finite simple groups (and their central extensions) that embed into exceptional Lie groups were classified across many papers. There is a summary by Griess and Ryba here. J1 does not appear to be one of them. But incidentally following Victor's suggestion I got Sym^4(56)^{J1} is 8-dimensional, for both the 56-dimensional representations.

"If things had happened normally, when the shepherds returned the envelope would have been broken open (we do not know why it remained intact) and the number of sheep and goats would have been checked by matching them with the clay balls. There could be no dispute, because two records had been kept: the clay balls for the shepherds, the seal and inscription for the owner."

and a little later from the same book, now describing an archaeologist's discovery: "...without knowing it, the uneducated servant had repeated a procedure used by equally uneducated shepherds who lived in that region 3500 years earlier. The clay envelope had belonged to an accountant (who, unlike the shepherds, knew how to write). The shepherds had gone to see him before taking their master's flock to pasture. He made as many clay balls as there were animals in the flock and put them into the envelope, which was then closed."

I think the story is Georges Ifrah's, available in English as From 1 to 0. "Let us imagine a shepherd, unable to count, who has a flock of sheep that he keeps in a cave. There are 55 of them, but he has no understanding of what "the number 55" means. He would like to be sure that all of them come back every evening. One day he has an idea. He sits down at the entrance of his cave and has the sheep go in one by one. Each time one of them passes, he makes a notch in a bone. When all the sheep have passed, he has made exactly 55 notches, without knowing the arithmetic meaning of that number."

On a circle the probability that a Brownian loop has length ell (is homotopic to a geodesic with length ell) is more like exp(-ell^2).

In your examples, the H_I might be off by a central extension. E.g. PGL(a) x PGL(b) is not usually a subgroup of PGL(a+b).

If your goal is to find examples with "lots" of torsion in H^3, instead of a specific group, Reznikov showed the twistor bundle over a hyperbolic 4-manifold is symplectic.

Complex K-theory? A 3-manifold has K^0 = H^0 + H^2 and K^1 = H^1 + H^3. Looks like Z^{2g+1} + Z/(2g -2) in degree zero and Z^{2g+1} in degree 1.