\begin{align} &f(x)=x^5+x^4+2x^3+4x^2+x+1 \notag \\ &Galois \ Resolvent : V(v) \equiv v^{120}+360v^{119}+64740v^{118}+.... =\displaystyle \prod_{i=1}^{6}V_{i}(v)\notag \\ \end{align}

\begin{align} V_{1}&={{v}^{20}}+60 {{v}^{19}}+1790 {{v}^{18}}+35100 {{v}^{17}}+505261 {{v}^{16}}+5652840 {{v}^{15}}+50799180 {{v}^{14}}+373971600 {{v}^{13}} \notag \\ &+2281089966 {{v}^{12}}+11590327440 {{v}^{11}}+49084357780 {{v}^{10}}+172620188400 {{v}^{9}}+500267581306 {{v}^{8}} \notag \\ &+1181164237800 {{v}^{7}}+2242276888380{{v}^{6}}+3401909638560 {{v}^{5}}+4254933143241 {{v}^{4}} \notag \\ &+4933817387460 {{v}^{3}}+6084439227750 {{v}^{2}}+7164705190500 v+5919446204113 \notag \\ \end{align}

\begin{align} &Galois \ Group \ and \ its \ Composition \ Series \notag \\ \end{align} \begin{align} &\quad & F_{20}&=\bigl[\ 1,8,18,23,30,33,40,43,52,59,61,70,73,80,90,95,99,108,110,117 \ \bigr] \notag \\ &\quad & D_{5}&=\bigl[ \ 1,8,30,43,52,61,90,95,108,117 \ \bigr] \notag \\ &\quad & C_{5}&= \bbox[#FFFF00]{\bigl[ \ 43,117,90,52,1 \ \bigr]} \quad \Leftarrow \quad \bbox[#FFFF00]{ \bigl\{ \rho_{43}^2=\rho_{117}, \ \rho_{43}^3=\rho_{90}, \ \rho_{43}^4=\rho_{52}, \ \rho_{43}^5=\rho_{1} \bigr\} }\notag \\ &\quad & e&=\bigl[ \ 1 \ \bigr] \notag \\ \end{align}

\begin{align} &Conjugacy \ Classes \ of \ F_{20} \notag \\ \end{align} \begin{align} &\quad & \varLambda_1&=[1] & \varLambda_2&=\bbox[#FFFF00]{[43,117,90,52]} & \varLambda_3&=[8,30,61,95,108] \notag \\ &\quad & \varLambda_4&=[18,33,70,80,99] & \varLambda_5&=[23,40,59,73,110] & & \notag \\ \end{align}

\begin{align} \bbox[#FFFF00]{ \biggl[ \ F_{20}= \varLambda_1+\varLambda_2+\varLambda_3+\varLambda_4+\varLambda_5 \quad \triangleright \quad D_{5}=\varLambda_1+\varLambda_2+\varLambda_3 \quad \triangleright \quad C_{5}=\varLambda_1+\varLambda_2 \quad \triangleright \quad e=\varLambda_1 \ \biggr] } \notag \\ \end{align}

\begin{align} V_{2}&={{v}^{20}}+60 {{v}^{19}}+1790 {{v}^{18}}+35100 {{v}^{17}}+505261 {{v}^{16}}+5655960 {{v}^{15}}+50923980 {{v}^{14}}+376545600 {{v}^{13}} \notag \\ &+2317001166 {{v}^{12}}+11964256320 {{v}^{11}}+52107272740 {{v}^{10}}+191853693600 {{v}^{9}}+596962688906 {{v}^{8}} \notag \\ &+1566271269720 {{v}^{7}}+3458738963260 {{v}^{6}}+6423623108400 {{v}^{5}}+9958500810441 {{v}^{4}} \notag \\ &+12461335397940 {{v}^{3}}+11642922808790 {{v}^{2}}+7043489742900 v+2039290237153 \notag \\ \end{align}

\begin{align} &Galois \ Group \ and \ its \ Composition \ Series \notag \\ \end{align} \begin{align} &\quad & F_{20}&=\bigl[\ 1,10,17,19,26,36,37,47,51,59,64,68,78,80,87,96,98,108,109,119 \ \bigr] \notag \\ &\quad & D_{5}&=\bigl[ \ 1,17,26,47,64,68,78,87,108,109 \ \bigr] \notag \\ &\quad & C_{5}&=\bigl[ \ 47,109,64,78,1 \ \bigr] \notag \\ &\quad & e&=\bigl[ \ 1 \ \bigr] \notag \\ \end{align}

\begin{align} &Conjugacy \ Classes \ of \ F_{20} \notag \\ \end{align} \begin{align} &\quad & \varLambda_1&=[1] & \varLambda_2&=[47,109,64,78] & \varLambda_3&=[17,26,68,87,108] \notag \\ &\quad & \varLambda_4&=[10,37,59,96,98] & \varLambda_5&=[19,36,51,80,119] & & \notag \\ \end{align}

\begin{align} V_{3}&={{v}^{20}}+60 {{v}^{19}}+1790 {{v}^{18}}+35100 {{v}^{17}}+505261 {{v}^{16}}+5657520 {{v}^{15}}+51001980 {{v}^{14}}+378300600 {{v}^{13}} \notag \\ &+2341243566 {{v}^{12}}+12196523160 {{v}^{11}}+53761666780 {{v}^{10}}+201106997400 {{v}^{9}}+639709411106 {{v}^{8}} \notag \\ &+1735342823280 {{v}^{7}}+4035147522220 {{v}^{6}}+8090223429240 {{v}^{5}}+13945832028441 {{v}^{4}} \notag \\ &+20035467005580 {{v}^{3}}+22125472711550 {{v}^{2}}+16079157042780 v+6837132314233 \notag \\ \end{align}

\begin{align} &Galois \ Group \ and \ its \ Composition \ Series \notag \\ \end{align} \begin{align} &\quad & F_{20}&=\bigl[\ 1,8,18,23,27,36,38,45,53,58,63,68,76,83,85,94,98,103,113,120 \ \bigr] \notag \\ &\quad & D_{5}&=\bigl[ \ 1,8,27,38,53,68,83,94,113,120 \ \bigr] \notag \\ &\quad & C_{5}&=\bigl[ \ 38,94,113,53,1 \ \bigr] \notag \\ &\quad & e&=\bigl[ \ 1 \ \bigr] \notag \\ \end{align}

\begin{align} &Conjugacy \ Classes \ of \ F_{20} \notag \\ \end{align} \begin{align} &\quad & \varLambda_1&=[1] & \varLambda_2&=[38,94,113,53] & \varLambda_3&=[8,27,68,83,120] \notag \\ &\quad & \varLambda_4&=[18,45,58,85,98] & \varLambda_5&=[23,36,63,76,103] & & \notag \\ \end{align}

\begin{align} V_{4}&={{v}^{20}}+60 {{v}^{19}}+1790 {{v}^{18}}+35100 {{v}^{17}}+505261 {{v}^{16}}+5657520 {{v}^{15}}+51033180 {{v}^{14}}+379611000 {{v}^{13}} \notag \\ &+2367283086 {{v}^{12}}+12520678680 {{v}^{11}}+56578689820 {{v}^{10}}+219122002200 {{v}^{9}}+726542093666 {{v}^{8}} \notag \\ &+2051322345840 {{v}^{7}}+4880224987660 {{v}^{6}}+9607234534200 {{v}^{5}}+15166159082361 {{v}^{4}} \notag \\ &+ 18160354917900 {{v}^{3}}+14980013767550 {{v}^{2}}+7296354097500 v+1761063045625 \notag \\ \end{align}

\begin{align} &Galois \ Group \ and \ its \ Composition \ Series \notag \\ \end{align} \begin{align} &\quad & F_{20}&=\bigl[\ 1,11,14,24,30,34,37,45,51,56,65,70,76,84,87,91,97,107,110,120 \ \bigr] \notag \\ &\quad & D_{5}&=\bigl[ \ 1,24,30,34,56,65,87,91,97,120 \ \bigr] \notag \\ &\quad & C_{5}&=\bigl[ \ 34,65,91,97,1 \ \bigr] \notag \\ &\quad & e&=\bigl[ \ 1 \ \bigr] \notag \\ \end{align}

\begin{align} &Conjugacy \ Classes \ of \ F_{20} \notag \\ \end{align} \begin{align} &\quad & \varLambda_1&=[1] & \varLambda_2&=[34,65,91,97] & \varLambda_3&=[24,30,56,87,120] \notag \\ &\quad & \varLambda_4&=[11,37,70,76,107] & \varLambda_5&=[14,45,51,84,110] & & \notag \\ \end{align}

\begin{align} V_{5}&={{v}^{20}}+60 {{v}^{19}}+1790 {{v}^{18}}+35100 {{v}^{17}}+505261 {{v}^{16}}+5659080 {{v}^{15}}+51064380 {{v}^{14}}+379400400 {{v}^{13}} \notag \\ &+2351679966 {{v}^{12}}+12238407600 {{v}^{11}}+53492805460 {{v}^{10}}+195251014800 {{v}^{9}}+587807064506 {{v}^{8}} \notag \\ &+1431313551240 {{v}^{7}}+2742285017260 {{v}^{6}}+3974293793760 {{v}^{5}}+4088970374841 {{v}^{4}} \notag \\ &+2626911907620 {{v}^{3}}+675718360070 {{v}^{2}}-213666065340 v+15823731313 \notag \\ \end{align}

\begin{align} &Galois \ Group \ and \ its \ Composition \ Series \notag \\ \end{align} \begin{align} &\quad & F_{20}&=\bigl[\ 1,11,14,24,26,33,42,44,54,58,61,69,73,83,86,96,101,103,112,119 \ \bigr] \notag \\ &\quad & D_{5}&=\bigl[ \ 1,24,26,42,61,69,83,86,101,112 \ \bigr] \notag \\ &\quad & C_{5}&=\bigl[ \ 42,86,69,101,1 \ \bigr] \notag \\ &\quad & e&=\bigl[ \ 1 \ \bigr] \notag \\ \end{align}

\begin{align} &Conjugacy \ Classes \ of \ F_{20} \notag \\ \end{align} \begin{align} &\quad & \varLambda_1&=[1] & \varLambda_2&=[42,86,69,101] & \varLambda_3&=[24,26,61,83,112] \notag \\ &\quad & \varLambda_4&=[11,44,58,73,119] & \varLambda_5&=[14,33,54,96,103] & & \notag \\ \end{align}

\begin{align} V_{6}&={{v}^{20}}+60 {{v}^{19}}+1790 {{v}^{18}}+35100 {{v}^{17}}+505261 {{v}^{16}}+5662200 {{v}^{15}}+51220380 {{v}^{14}}+383284800 {{v}^{13}} \notag \\ &+2414329566 {{v}^{12}}+12961651680{{v}^{11}}+59789211940 {{v}^{10}}+237890884800 {{v}^{9}}+816378602506 {{v}^{8}} \notag \\ &+2407592691960 {{v}^{7}}+6059842042380 {{v}^{6}}+12880967649840 {{v}^{5}}+22758667394841 {{v}^{4}} \notag \\ &+32586900570900 {{v}^{3}}+36272321795190 {{v}^{2}}+28893115997460 v+13542366738433 \notag \\ \end{align}

\begin{align} &Galois \ Group \ and \ its \ Composition \ Series \notag \\ \end{align} \begin{align} &\quad & F_{20}&=\bigl[\ 1,10,17,19,27,35,40,44,54,56,63,72,74,84,85,95,99,107,112,116 \ \bigr] \notag \\ &\quad & D_{5}&=\bigl[ \ 1,17,27,35,56,72,74,95,112,116 \ \bigr] \notag \\ &\quad & C_{5}&=\bigl[ \ 35,72,116,74,1 \ \bigr] \notag \\ &\quad & e&=\bigl[ \ 1 \ \bigr] \notag \\ \end{align}

\begin{align} &Conjugacy \ Classes \ of \ F_{20} \notag \\ \end{align} \begin{align} &\quad & \varLambda_1&=[1] & \varLambda_2&=[35,72,116,74] & \varLambda_3&=[17,27,56,95,112] \notag \\ &\quad & \varLambda_4&=[10,44,63,84,99] & \varLambda_5&=[19,40,54,85,107] & & \notag \\ \end{align}