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Elements of the group are displayed as words in the presentation $\langle a, b, c, d, e, f, g, h \mid a^{8}=b^{3}=c^{3}=d^{3}=e^{3}=f^{3}=g^{3}=h^{3}=[b,d]=[b,f]=[b,g]=[b,h]=[c,d]=[c,f]=[c,h]=[d,e]=[d,f]=[d,g]=[d,h]=[e,f]=[e,h]=[f,g]=[f,h]=[g,h]=1, b^{a}=bdfg^{2}h^{2}, c^{a}=ce^{2}fg^{2}h, d^{a}=df^{2}h, e^{a}=bc^{2}fh, f^{a}=d^{2}f, g^{a}=b^{2}d^{2}fh, h^{a}=d^{2}f^{2}, c^{b}=cd^{2}f, e^{b}=eh, e^{c}=d^{2}eh^{2}, g^{c}=d^{2}g, g^{e}=f^{2}g \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P
$C_3^5:F_9$ 1A $1$ $1$ $C_3^5:F_9$ 1A 1A $1$
$C_3^5:F_9$ 2A $2$ $81$ $C_3^3:C_8$ 1A 2A $a^{4}c^{2}e^{2}g^{2}h^{2}$
$C_3^5:F_9$ 3A $3$ $2$ $C_3^5.C_3^2.C_4$ 3A 1A $d^{2}h$
$C_3^5:F_9$ 3B $3$ $4$ $C_3^4.C_3^3.C_2$ 3B 1A $d^{2}f^{2}$
$C_3^5:F_9$ 3C $3$ $4$ $C_3^4.C_3^3.C_2$ 3C 1A $d^{2}f^{2}h$
$C_3^5:F_9$ 3D1 $3$ $4$ $C_3^4.C_3^3.C_2$ 3D-1 1A $f^{2}$
$C_3^5:F_9$ 3D-1 $3$ $4$ $C_3^4.C_3^3.C_2$ 3D1 1A $f$
$C_3^5:F_9$ 3E1 $3$ $4$ $C_3^4.C_3^3.C_2$ 3E-1 1A $fh$
$C_3^5:F_9$ 3E-1 $3$ $4$ $C_3^4.C_3^3.C_2$ 3E1 1A $f^{2}h^{2}$
$C_3^5:F_9$ 3F $3$ $72$ $C_3^5$ 3F 1A $g$
$C_3^5:F_9$ 3G $3$ $72$ $C_3^5$ 3G 1A $ce$
$C_3^5:F_9$ 3H $3$ $72$ $C_3^5$ 3H 1A $be$
$C_3^5:F_9$ 3I $3$ $72$ $C_3^5$ 3I 1A $bc$
$C_3^5:F_9$ 3J1 $3$ $72$ $C_3^5$ 3J-1 1A $gh$
$C_3^5:F_9$ 3J-1 $3$ $72$ $C_3^5$ 3J1 1A $gh^{2}$
$C_3^5:F_9$ 3K1 $3$ $72$ $C_3^5$ 3K-1 1A $ceh$
$C_3^5:F_9$ 3K-1 $3$ $72$ $C_3^5$ 3K1 1A $cde$
$C_3^5:F_9$ 3L1 $3$ $72$ $C_3^5$ 3L-1 1A $bde$
$C_3^5:F_9$ 3L-1 $3$ $72$ $C_3^5$ 3L1 1A $bd^{2}e$
$C_3^5:F_9$ 3M1 $3$ $72$ $C_3^5$ 3M-1 1A $bch$
$C_3^5:F_9$ 3M-1 $3$ $72$ $C_3^5$ 3M1 1A $bch^{2}$
$C_3^5:F_9$ 3N $3$ $216$ $C_3^4$ 3N 1A $cde^{2}f^{2}g^{2}h$
$C_3^5:F_9$ 3O $3$ $216$ $C_3^4$ 3O 1A $bcde^{2}fg^{2}$
$C_3^5:F_9$ 3P $3$ $216$ $C_3^4$ 3P 1A $cde^{2}$
$C_3^5:F_9$ 3Q $3$ $216$ $C_3^4$ 3Q 1A $bcde^{2}fh$
$C_3^5:F_9$ 3R $3$ $216$ $C_3^4$ 3R 1A $bc^{2}d^{2}ef^{2}h$
$C_3^5:F_9$ 3S $3$ $216$ $C_3^4$ 3S 1A $bcdeh$
$C_3^5:F_9$ 4A1 $4$ $729$ $C_3:C_8$ 2A 4A-1 $a^{6}c^{2}defgh$
$C_3^5:F_9$ 4A-1 $4$ $729$ $C_3:C_8$ 2A 4A1 $a^{2}deg$
$C_3^5:F_9$ 6A $6$ $162$ $C_3^2:C_{12}$ 3A 2A $a^{4}ce^{2}fg^{2}$
$C_3^5:F_9$ 6B $6$ $324$ $C_3^2\times C_6$ 3B 2A $a^{4}c^{2}def^{2}g$
$C_3^5:F_9$ 6C $6$ $324$ $C_3^2\times C_6$ 3C 2A $a^{4}c^{2}df^{2}$
$C_3^5:F_9$ 6D1 $6$ $324$ $C_3^2\times C_6$ 3D1 2A $a^{4}b^{2}c^{2}d^{2}eg^{2}$
$C_3^5:F_9$ 6D-1 $6$ $324$ $C_3^2\times C_6$ 3D-1 2A $a^{4}b^{2}c^{2}d^{2}efg^{2}$
$C_3^5:F_9$ 6E1 $6$ $324$ $C_3^2\times C_6$ 3E1 2A $a^{4}b^{2}cdf^{2}g^{2}$
$C_3^5:F_9$ 6E-1 $6$ $324$ $C_3^2\times C_6$ 3E-1 2A $a^{4}b^{2}cdfg^{2}h^{2}$
$C_3^5:F_9$ 8A1 $8$ $2187$ $C_8$ 4A1 8A3 $a^{7}c^{2}d$
$C_3^5:F_9$ 8A-1 $8$ $2187$ $C_8$ 4A-1 8A-3 $acd^{2}e^{2}f^{2}g^{2}$
$C_3^5:F_9$ 8A3 $8$ $2187$ $C_8$ 4A-1 8A1 $a^{5}cd^{2}h^{2}$
$C_3^5:F_9$ 8A-3 $8$ $2187$ $C_8$ 4A1 8A-1 $a^{3}de^{2}fg^{2}$
$C_3^5:F_9$ 12A1 $12$ $1458$ $C_{12}$ 6A 4A1 $a^{2}c^{2}e^{2}g^{2}h^{2}$
$C_3^5:F_9$ 12A-1 $12$ $1458$ $C_{12}$ 6A 4A-1 $a^{6}c^{2}fh$
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