Here's the factorization (over Q) of the groebner basis of the radical ideal. I've introduced extra variables I call bb,cc,dd,ee,ff which are the inverses of b,c,d,e,f respectively, enforced by equations like b*bb-1==0. The solutions are obtained by setting all of these polynomials to 0. Victor [f*ff - 1, e*ee - 1, d*dd - 1, c*cc - 1, b*bb - 1, b*c*d*e*f^2 + b^2*c*d*f + b*c^2*e*f + b*c*d*e*f + b*d^2*e*f + c*d*e^2*f + b*c*d*e, b*c*d*e^2*f + b^2*c*d*e + b*c*d^2*f + b*c*d*e*f + c^2*d*e*f + b*d*e*f^2 + b*c*e*f, b*c*d^2*e*f + b*c^2*d*e + b^2*d*e*f + b*c*d*e*f + b*c*e^2*f + c*d*e*f^2 + b*c*d*f, b*c^2*d*e*f + b*c*d^2*e + b^2*c*e*f + b*c*d*e*f + b*d*e^2*f + b*c*d*f^2 + c*d*e*f, b^2*c*d*e*f + b*c*d*e^2 + b*c^2*d*f + b*c*d*e*f + c*d^2*e*f + b*c*e*f^2 + b*d*e*f, (ff - 1) * (ff^2 - ff + 1) * (ff^2 + ff + 1) * (ff^2 + 5*ff + 1) * (5*ff^2 - 17*ff + 5) * (5*ff^2 + 11*ff + 5) * (ff^6 + ff^5 + ff^4 + ff^3 + ff^2 + ff + 1) * (4*ff^6 + 4*ff^5 + 32*ff^4 - 17*ff^3 + 32*ff^2 + 4*ff + 4) * (4*ff^18 + 12*ff^17 + 162*ff^16 + 321*ff^15 - 288*ff^14 - 1866*ff^13 - 4056*ff^12 - 6618*ff^11 - 8362*ff^10 - 8794*ff^9 - 8362*ff^8 - 6618*ff^7 - 4056*ff^6 - 1866*ff^5 - 288*ff^4 + 321*ff^3 + 162*ff^2 + 12*ff + 4), (f - 1) * (f^2 - f + 1) * (f^2 + f + 1) * (f^2 + 5*f + 1) * (5*f^2 - 17*f + 5) * (5*f^2 + 11*f + 5) * (f^6 + f^5 + f^4 + f^3 + f^2 + f + 1) * (4*f^6 + 4*f^5 + 32*f^4 - 17*f^3 + 32*f^2 + 4*f + 4) * (4*f^18 + 12*f^17 + 162*f^16 + 321*f^15 - 288*f^14 - 1866*f^13 - 4056*f^12 - 6618*f^11 - 8362*f^10 - 8794*f^9 - 8362*f^8 - 6618*f^7 - 4056*f^6 - 1866*f^5 - 288*f^4 + 321*f^3 + 162*f^2 + 12*f + 4), (ee - 1) * (ee + 1) * (ee^2 + ee + 1) * (ee^2 + 5*ee + 1) * (2*ee^2 + 3*ee + 2) * (ee^6 + ee^5 - 6*ee^4 - 13*ee^3 - 6*ee^2 + ee + 1) * (ee^6 + ee^5 + ee^4 + ee^3 + ee^2 + ee + 1) * (36*ee^8 - 36*ee^7 + 378*ee^6 + 294*ee^5 + 1057*ee^4 + 294*ee^3 + 378*ee^2 - 36*ee + 36) * (16*ee^36 + 96*ee^35 + 796*ee^34 + 1804*ee^33 + 17566*ee^32 + 38418*ee^31 + 165145*ee^30 + 451622*ee^29 + 1275250*ee^28 + 2800516*ee^27 + 6052848*ee^26 + 11530332*ee^25 + 19573592*ee^24 + 29211804*ee^23 + 39747080*ee^22 + 49794120*ee^21 + 58055092*ee^20 + 63387800*ee^19 + 65199950*ee^18 + 63387800*ee^17 + 58055092*ee^16 + 49794120*ee^15 + 39747080*ee^14 + 29211804*ee^13 + 19573592*ee^12 + 11530332*ee^11 + 6052848*ee^10 + 2800516*ee^9 + 1275250*ee^8 + 451622*ee^7 + 165145*ee^6 + 38418*ee^5 + 17566*ee^4 + 1804*ee^3 + 796*ee^2 + 96*ee + 16), (dd - 1) * (dd + 1) * (dd^2 + dd + 1) * (dd^2 + 5*dd + 1) * (2*dd^2 + 3*dd + 2) * (dd^6 + dd^5 - 6*dd^4 - 13*dd^3 - 6*dd^2 + dd + 1) * (dd^6 + dd^5 + dd^4 + dd^3 + dd^2 + dd + 1) * (36*dd^8 - 36*dd^7 + 378*dd^6 + 294*dd^5 + 1057*dd^4 + 294*dd^3 + 378*dd^2 - 36*dd + 36) * (16*dd^36 + 96*dd^35 + 796*dd^34 + 1804*dd^33 + 17566*dd^32 + 38418*dd^31 + 165145*dd^30 + 451622*dd^29 + 1275250*dd^28 + 2800516*dd^27 + 6052848*dd^26 + 11530332*dd^25 + 19573592*dd^24 + 29211804*dd^23 + 39747080*dd^22 + 49794120*dd^21 + 58055092*dd^20 + 63387800*dd^19 + 65199950*dd^18 + 63387800*dd^17 + 58055092*dd^16 + 49794120*dd^15 + 39747080*dd^14 + 29211804*dd^13 + 19573592*dd^12 + 11530332*dd^11 + 6052848*dd^10 + 2800516*dd^9 + 1275250*dd^8 + 451622*dd^7 + 165145*dd^6 + 38418*dd^5 + 17566*dd^4 + 1804*dd^3 + 796*dd^2 + 96*dd + 16), (e - 1) * (e + 1) * (e^2 + e + 1) * (e^2 + 5*e + 1) * (2*e^2 + 3*e + 2) * (e^6 + e^5 - 6*e^4 - 13*e^3 - 6*e^2 + e + 1) * (e^6 + e^5 + e^4 + e^3 + e^2 + e + 1) * (36*e^8 - 36*e^7 + 378*e^6 + 294*e^5 + 1057*e^4 + 294*e^3 + 378*e^2 - 36*e + 36) * (16*e^36 + 96*e^35 + 796*e^34 + 1804*e^33 + 17566*e^32 + 38418*e^31 + 165145*e^30 + 451622*e^29 + 1275250*e^28 + 2800516*e^27 + 6052848*e^26 + 11530332*e^25 + 19573592*e^24 + 29211804*e^23 + 39747080*e^22 + 49794120*e^21 + 58055092*e^20 + 63387800*e^19 + 65199950*e^18 + 63387800*e^17 + 58055092*e^16 + 49794120*e^15 + 39747080*e^14 + 29211804*e^13 + 19573592*e^12 + 11530332*e^11 + 6052848*e^10 + 2800516*e^9 + 1275250*e^8 + 451622*e^7 + 165145*e^6 + 38418*e^5 + 17566*e^4 + 1804*e^3 + 796*e^2 + 96*e + 16), (d - 1) * (d + 1) * (d^2 + d + 1) * (d^2 + 5*d + 1) * (2*d^2 + 3*d + 2) * (d^6 + d^5 - 6*d^4 - 13*d^3 - 6*d^2 + d + 1) * (d^6 + d^5 + d^4 + d^3 + d^2 + d + 1) * (36*d^8 - 36*d^7 + 378*d^6 + 294*d^5 + 1057*d^4 + 294*d^3 + 378*d^2 - 36*d + 36) * (16*d^36 + 96*d^35 + 796*d^34 + 1804*d^33 + 17566*d^32 + 38418*d^31 + 165145*d^30 + 451622*d^29 + 1275250*d^28 + 2800516*d^27 + 6052848*d^26 + 11530332*d^25 + 19573592*d^24 + 29211804*d^23 + 39747080*d^22 + 49794120*d^21 + 58055092*d^20 + 63387800*d^19 + 65199950*d^18 + 63387800*d^17 + 58055092*d^16 + 49794120*d^15 + 39747080*d^14 + 29211804*d^13 + 19573592*d^12 + 11530332*d^11 + 6052848*d^10 + 2800516*d^9 + 1275250*d^8 + 451622*d^7 + 165145*d^6 + 38418*d^5 + 17566*d^4 + 1804*d^3 + 796*d^2 + 96*d + 16), (cc - 1) * (cc^2 - cc + 1) * (cc^2 + cc + 1) * (cc^2 + 5*cc + 1) * (2*cc^2 + 3*cc + 2) * (5*cc^2 - 17*cc + 5) * (5*cc^2 + 11*cc + 5) * (cc^6 + cc^5 + cc^4 + cc^3 + cc^2 + cc + 1) * (4*cc^6 + 4*cc^5 + 18*cc^4 + 11*cc^3 + 18*cc^2 + 4*cc + 4) * (36*cc^8 - 36*cc^7 + 378*cc^6 + 294*cc^5 + 1057*cc^4 + 294*cc^3 + 378*cc^2 - 36*cc + 36) * (256*cc^36 + 1536*cc^35 + 3648*cc^34 - 7232*cc^33 - 21824*cc^32 + 76448*cc^31 + 121232*cc^30 - 418008*cc^29 + 454380*cc^28 + 457840*cc^27 - 1045658*cc^26 + 226204*cc^25 + 1449103*cc^24 - 1215674*cc^23 - 647357*cc^22 + 1855998*cc^21 - 514038*cc^20 - 796110*cc^19 + 1308641*cc^18 - 796110*cc^17 - 514038*cc^16 + 1855998*cc^15 - 647357*cc^14 - 1215674*cc^13 + 1449103*cc^12 + 226204*cc^11 - 1045658*cc^10 + 457840*cc^9 + 454380*cc^8 - 418008*cc^7 + 121232*cc^6 + 76448*cc^5 - 21824*cc^4 - 7232*cc^3 + 3648*cc^2 + 1536*cc + 256), (bb - 1) * (bb^2 - bb + 1) * (bb^2 + bb + 1) * (bb^2 + 5*bb + 1) * (2*bb^2 + 3*bb + 2) * (5*bb^2 - 17*bb + 5) * (5*bb^2 + 11*bb + 5) * (bb^6 + bb^5 + bb^4 + bb^3 + bb^2 + bb + 1) * (4*bb^6 + 4*bb^5 + 18*bb^4 + 11*bb^3 + 18*bb^2 + 4*bb + 4) * (36*bb^8 - 36*bb^7 + 378*bb^6 + 294*bb^5 + 1057*bb^4 + 294*bb^3 + 378*bb^2 - 36*bb + 36) * (256*bb^36 + 1536*bb^35 + 3648*bb^34 - 7232*bb^33 - 21824*bb^32 + 76448*bb^31 + 121232*bb^30 - 418008*bb^29 + 454380*bb^28 + 457840*bb^27 - 1045658*bb^26 + 226204*bb^25 + 1449103*bb^24 - 1215674*bb^23 - 647357*bb^22 + 1855998*bb^21 - 514038*bb^20 - 796110*bb^19 + 1308641*bb^18 - 796110*bb^17 - 514038*bb^16 + 1855998*bb^15 - 647357*bb^14 - 1215674*bb^13 + 1449103*bb^12 + 226204*bb^11 - 1045658*bb^10 + 457840*bb^9 + 454380*bb^8 - 418008*bb^7 + 121232*bb^6 + 76448*bb^5 - 21824*bb^4 - 7232*bb^3 + 3648*bb^2 + 1536*bb + 256), (c - 1) * (c^2 - c + 1) * (c^2 + c + 1) * (c^2 + 5*c + 1) * (2*c^2 + 3*c + 2) * (5*c^2 - 17*c + 5) * (5*c^2 + 11*c + 5) * (c^6 + c^5 + c^4 + c^3 + c^2 + c + 1) * (4*c^6 + 4*c^5 + 18*c^4 + 11*c^3 + 18*c^2 + 4*c + 4) * (36*c^8 - 36*c^7 + 378*c^6 + 294*c^5 + 1057*c^4 + 294*c^3 + 378*c^2 - 36*c + 36) * (256*c^36 + 1536*c^35 + 3648*c^34 - 7232*c^33 - 21824*c^32 + 76448*c^31 + 121232*c^30 - 418008*c^29 + 454380*c^28 + 457840*c^27 - 1045658*c^26 + 226204*c^25 + 1449103*c^24 - 1215674*c^23 - 647357*c^22 + 1855998*c^21 - 514038*c^20 - 796110*c^19 + 1308641*c^18 - 796110*c^17 - 514038*c^16 + 1855998*c^15 - 647357*c^14 - 1215674*c^13 + 1449103*c^12 + 226204*c^11 - 1045658*c^10 + 457840*c^9 + 454380*c^8 - 418008*c^7 + 121232*c^6 + 76448*c^5 - 21824*c^4 - 7232*c^3 + 3648*c^2 + 1536*c + 256), (b - 1) * (b^2 - b + 1) * (b^2 + b + 1) * (b^2 + 5*b + 1) * (2*b^2 + 3*b + 2) * (5*b^2 - 17*b + 5) * (5*b^2 + 11*b + 5) * (b^6 + b^5 + b^4 + b^3 + b^2 + b + 1) * (4*b^6 + 4*b^5 + 18*b^4 + 11*b^3 + 18*b^2 + 4*b + 4) * (36*b^8 - 36*b^7 + 378*b^6 + 294*b^5 + 1057*b^4 + 294*b^3 + 378*b^2 - 36*b + 36) * (256*b^36 + 1536*b^35 + 3648*b^34 - 7232*b^33 - 21824*b^32 + 76448*b^31 + 121232*b^30 - 418008*b^29 + 454380*b^28 + 457840*b^27 - 1045658*b^26 + 226204*b^25 + 1449103*b^24 - 1215674*b^23 - 647357*b^22 + 1855998*b^21 - 514038*b^20 - 796110*b^19 + 1308641*b^18 - 796110*b^17 - 514038*b^16 + 1855998*b^15 - 647357*b^14 - 1215674*b^13 + 1449103*b^12 + 226204*b^11 - 1045658*b^10 + 457840*b^9 + 454380*b^8 - 418008*b^7 + 121232*b^6 + 76448*b^5 - 21824*b^4 - 7232*b^3 + 3648*b^2 + 1536*b + 256)]