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Elements of the group are displayed as words in the presentation $\langle a, b, c, d, e, f \mid b^{3}=d^{21}=e^{7}=f^{7}=[c,f]=[e,f]=1, a^{2}=cde^{2}f, c^{3}=f^{6}, b^{a}=b^{2}d^{3}e^{2}f^{5}, c^{a}=c^{2}d^{15}e^{4}f, d^{a}=c^{2}d^{7}e^{5}f^{2}, e^{a}=d^{9}f^{3}, f^{a}=f^{4}, c^{b}=c^{2}d^{4}e, d^{b}=c^{2}d^{9}e^{2}f^{6}, e^{b}=d^{18}e^{5}f^{5}, f^{b}=e^{2}f^{2}, d^{c}=d^{16}e^{2}, e^{c}=e^{4}f^{3}, e^{d}=e^{4}f^{2}, f^{d}=f^{2} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P 7P
$C_7^3:C_3^2:S_3$ 1A $1$ $1$ $C_7^3:C_3^2:S_3$ 1A 1A 1A $1$
$C_7^3:C_3^2:S_3$ 2A $2$ $63$ $C_7^2:C_6$ 1A 2A 2A $ad^{7}e^{5}f$
$C_7^3:C_3^2:S_3$ 3A $3$ $294$ $C_{21}:C_3$ 3A 1A 3A $bcd^{19}e^{3}f^{2}$
$C_7^3:C_3^2:S_3$ 3B $3$ $294$ $C_{21}:C_3$ 3B 1A 3B $bc^{2}d^{13}e^{5}f^{2}$
$C_7^3:C_3^2:S_3$ 3C $3$ $294$ $C_{21}:C_3$ 3C 1A 3C $bdf$
$C_7^3:C_3^2:S_3$ 3D $3$ $294$ $C_{21}:C_3$ 3D 1A 3D $d^{7}e^{6}$
$C_7^3:C_3^2:S_3$ 3E1 $3$ $343$ $C_3^2:S_3$ 3E-1 1A 3E1 $cd^{19}e^{4}f$
$C_7^3:C_3^2:S_3$ 3E-1 $3$ $343$ $C_3^2:S_3$ 3E1 1A 3E-1 $c^{2}d^{8}e^{6}$
$C_7^3:C_3^2:S_3$ 6A1 $6$ $3087$ $C_6$ 3E1 2A 6A1 $ac^{2}d^{15}e^{4}f$
$C_7^3:C_3^2:S_3$ 6A-1 $6$ $3087$ $C_6$ 3E-1 2A 6A-1 $acd^{5}e$
$C_7^3:C_3^2:S_3$ 7A1 $7$ $9$ $C_7\wr S_3$ 7A1 7A-1 1A $d^{3}e^{3}f^{2}$
$C_7^3:C_3^2:S_3$ 7A-1 $7$ $9$ $C_7\wr S_3$ 7A-1 7A1 1A $d^{18}e^{4}f^{5}$
$C_7^3:C_3^2:S_3$ 7B1 $7$ $9$ $C_7\wr S_3$ 7B1 7B-1 1A $d^{3}e^{5}f^{4}$
$C_7^3:C_3^2:S_3$ 7B-1 $7$ $9$ $C_7\wr S_3$ 7B-1 7B1 1A $d^{18}e^{2}f^{3}$
$C_7^3:C_3^2:S_3$ 7C1 $7$ $9$ $C_7\wr S_3$ 7C1 7C-1 1A $d^{12}e^{3}f^{6}$
$C_7^3:C_3^2:S_3$ 7C-1 $7$ $9$ $C_7\wr S_3$ 7C-1 7C1 1A $d^{9}e^{4}f$
$C_7^3:C_3^2:S_3$ 7D1 $7$ $9$ $C_7\wr S_3$ 7D1 7D-1 1A $d^{3}$
$C_7^3:C_3^2:S_3$ 7D-1 $7$ $9$ $C_7\wr S_3$ 7D-1 7D1 1A $d^{18}$
$C_7^3:C_3^2:S_3$ 7E1 $7$ $27$ $D_7\times C_7^2$ 7E1 7E-1 1A $d^{3}f^{5}$
$C_7^3:C_3^2:S_3$ 7E-1 $7$ $27$ $D_7\times C_7^2$ 7E-1 7E1 1A $d^{18}f^{2}$
$C_7^3:C_3^2:S_3$ 7F1 $7$ $27$ $D_7\times C_7^2$ 7F1 7F-1 1A $d^{9}e^{4}f^{2}$
$C_7^3:C_3^2:S_3$ 7F-1 $7$ $27$ $D_7\times C_7^2$ 7F-1 7F1 1A $d^{12}e^{3}f^{5}$
$C_7^3:C_3^2:S_3$ 7G1 $7$ $27$ $D_7\times C_7^2$ 7G1 7G-1 1A $d^{3}e^{3}f^{5}$
$C_7^3:C_3^2:S_3$ 7G-1 $7$ $27$ $D_7\times C_7^2$ 7G-1 7G1 1A $d^{18}e^{4}f^{2}$
$C_7^3:C_3^2:S_3$ 7H1 $7$ $27$ $D_7\times C_7^2$ 7H1 7H-1 1A $d^{3}ef^{6}$
$C_7^3:C_3^2:S_3$ 7H-1 $7$ $27$ $D_7\times C_7^2$ 7H-1 7H1 1A $d^{18}e^{6}f$
$C_7^3:C_3^2:S_3$ 7I $7$ $54$ $C_7^3$ 7I 7I 1A $e$
$C_7^3:C_3^2:S_3$ 14A1 $14$ $189$ $C_7\times C_{14}$ 7E1 14A-1 2A $ab^{2}c^{2}d^{18}e^{3}f^{6}$
$C_7^3:C_3^2:S_3$ 14A-1 $14$ $189$ $C_7\times C_{14}$ 7E-1 14A1 2A $ab^{2}c^{2}d^{15}e^{3}f$
$C_7^3:C_3^2:S_3$ 14B1 $14$ $189$ $C_7\times C_{14}$ 7D1 14B-1 2A $abcd^{7}e^{4}f$
$C_7^3:C_3^2:S_3$ 14B-1 $14$ $189$ $C_7\times C_{14}$ 7D-1 14B1 2A $abcd^{4}e^{4}f$
$C_7^3:C_3^2:S_3$ 14C1 $14$ $189$ $C_7\times C_{14}$ 7A-1 14C-1 2A $abcd^{11}f^{4}$
$C_7^3:C_3^2:S_3$ 14C-1 $14$ $189$ $C_7\times C_{14}$ 7A1 14C1 2A $abcd^{14}e^{5}$
$C_7^3:C_3^2:S_3$ 14D1 $14$ $189$ $C_7\times C_{14}$ 7F1 14D-1 2A $acd^{13}e^{3}f$
$C_7^3:C_3^2:S_3$ 14D-1 $14$ $189$ $C_7\times C_{14}$ 7F-1 14D1 2A $acd^{4}e^{6}f^{6}$
$C_7^3:C_3^2:S_3$ 14E1 $14$ $189$ $C_7\times C_{14}$ 7C1 14E-1 2A $abcd^{7}e^{3}f^{2}$
$C_7^3:C_3^2:S_3$ 14E-1 $14$ $189$ $C_7\times C_{14}$ 7C-1 14E1 2A $abcd^{4}f$
$C_7^3:C_3^2:S_3$ 14F1 $14$ $189$ $C_7\times C_{14}$ 7G1 14F-1 2A $ac^{2}d^{13}e^{4}$
$C_7^3:C_3^2:S_3$ 14F-1 $14$ $189$ $C_7\times C_{14}$ 7G-1 14F1 2A $ac^{2}d^{10}ef^{2}$
$C_7^3:C_3^2:S_3$ 14G1 $14$ $189$ $C_7\times C_{14}$ 7H1 14G-1 2A $ab^{2}c^{2}d^{18}f^{3}$
$C_7^3:C_3^2:S_3$ 14G-1 $14$ $189$ $C_7\times C_{14}$ 7H-1 14G1 2A $ab^{2}c^{2}d^{15}e^{6}f^{4}$
$C_7^3:C_3^2:S_3$ 14H1 $14$ $189$ $C_7\times C_{14}$ 7B-1 14H-1 2A $ab^{2}cdef^{3}$
$C_7^3:C_3^2:S_3$ 14H-1 $14$ $189$ $C_7\times C_{14}$ 7B1 14H1 2A $ab^{2}cd^{13}e^{6}f^{6}$
$C_7^3:C_3^2:S_3$ 21A1 $21$ $882$ $C_{21}$ 21A1 7A1 3A $bcd^{13}e^{4}f^{5}$
$C_7^3:C_3^2:S_3$ 21A-1 $21$ $882$ $C_{21}$ 21A-1 7A-1 3A $b^{2}c^{2}d^{8}e^{6}f$
$C_7^3:C_3^2:S_3$ 21B1 $21$ $882$ $C_{21}$ 21B1 7B1 3B $bc^{2}d^{7}e^{2}f$
$C_7^3:C_3^2:S_3$ 21B-1 $21$ $882$ $C_{21}$ 21B-1 7B-1 3B $b^{2}c^{2}e^{2}f^{6}$
$C_7^3:C_3^2:S_3$ 21C1 $21$ $882$ $C_{21}$ 21C1 7C1 3C $bd^{19}ef^{3}$
$C_7^3:C_3^2:S_3$ 21C-1 $21$ $882$ $C_{21}$ 21C-1 7C-1 3C $b^{2}c^{2}d^{13}ef^{2}$
$C_7^3:C_3^2:S_3$ 21D1 $21$ $882$ $C_{21}$ 21D1 7D-1 3D $d^{13}e^{6}$
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