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Elements of the group are displayed as words in the presentation $\langle a, b, c, d, e, f, g, h \mid b^{4}=c^{3}=d^{6}=e^{3}=f^{3}=g^{3}=h^{3}=[a,g]=[b,h]=[c,e]=[c,g]=[c,h]=[d,h]=[e,g]=[e,h]=[f,g]=[f,h]=[g,h]=1, a^{2}=d^{2}f^{2}g^{2}h^{2}, b^{a}=b^{3}cd^{2}f^{2}gh, c^{a}=c^{2}d^{4}fgh, d^{a}=df^{2}g^{2}h^{2}, e^{a}=e^{2}f^{2}h^{2}, f^{a}=fg^{2}h^{2}, h^{a}=h^{2}, c^{b}=c^{2}d^{4}fg^{2}, d^{b}=cde^{2}gh, e^{b}=e^{2}f^{2}g^{2}h^{2}, f^{b}=e^{2}f, g^{b}=g^{2}, d^{c}=deg^{2}h, f^{c}=fgh, e^{d}=e^{2}gh^{2}, f^{d}=f^{2}gh, g^{d}=g^{2}, f^{e}=fh^{2} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P
$\He_3^2.(C_2\times D_4)$ 1A $1$ $1$ $\He_3^2.(C_2\times D_4)$ 1A 1A $1$
$\He_3^2.(C_2\times D_4)$ 2A $2$ $27$ $C_2\times \He_3:D_4$ 1A 2A $d^{3}$
$\He_3^2.(C_2\times D_4)$ 2B $2$ $27$ $C_2\times \He_3:D_4$ 1A 2B $b^{2}d$
$\He_3^2.(C_2\times D_4)$ 2C $2$ $54$ $C_6.S_3^2$ 1A 2C $acg$
$\He_3^2.(C_2\times D_4)$ 2D $2$ $54$ $C_6.S_3^2$ 1A 2D $abd^{5}f^{2}g$
$\He_3^2.(C_2\times D_4)$ 2E $2$ $81$ $S_3\times D_{12}$ 1A 2E $b^{2}c^{2}d^{4}fh^{2}$
$\He_3^2.(C_2\times D_4)$ 2F $2$ $162$ $S_3\times D_6$ 1A 2F $ac^{2}defg^{2}h$
$\He_3^2.(C_2\times D_4)$ 2G $2$ $162$ $S_3\times D_6$ 1A 2G $ab^{3}cd^{4}ef^{2}gh^{2}$
$\He_3^2.(C_2\times D_4)$ 3A $3$ $2$ $C_3^2.C_3^4.C_4.C_2$ 3A 1A $h^{2}$
$\He_3^2.(C_2\times D_4)$ 3B $3$ $2$ $C_3^2.C_3^4.D_4$ 3B 1A $g$
$\He_3^2.(C_2\times D_4)$ 3C $3$ $4$ $C_3^2.C_3^4.C_4$ 3C 1A $g^{2}h^{2}$
$\He_3^2.(C_2\times D_4)$ 3D $3$ $24$ $C_3^4:C_6$ 3D 1A $c^{2}d^{2}e^{2}f^{2}g^{2}$
$\He_3^2.(C_2\times D_4)$ 3E $3$ $24$ $C_3^4:C_6$ 3E 1A $ce^{2}$
$\He_3^2.(C_2\times D_4)$ 3F $3$ $36$ $C_3\wr C_2^2$ 3F 1A $e^{2}fh$
$\He_3^2.(C_2\times D_4)$ 3G $3$ $36$ $C_3\wr C_2^2$ 3G 1A $egh^{2}$
$\He_3^2.(C_2\times D_4)$ 3H $3$ $36$ $C_3\wr C_2^2$ 3H 1A $c^{2}e^{2}gh^{2}$
$\He_3^2.(C_2\times D_4)$ 3I $3$ $36$ $C_3\wr C_2^2$ 3I 1A $c^{2}d^{2}e^{2}g$
$\He_3^2.(C_2\times D_4)$ 3J $3$ $48$ $C_3^2\times \He_3$ 3J 1A $ch$
$\He_3^2.(C_2\times D_4)$ 3K $3$ $48$ $C_3^2\times \He_3$ 3K 1A $d^{2}ef^{2}$
$\He_3^2.(C_2\times D_4)$ 3L $3$ $72$ $C_3^2\wr C_2$ 3L 1A $cd^{4}e^{2}f^{2}gh^{2}$
$\He_3^2.(C_2\times D_4)$ 3M $3$ $72$ $C_3^2\wr C_2$ 3M 1A $c^{2}fh$
$\He_3^2.(C_2\times D_4)$ 3N $3$ $144$ $C_3^4$ 3N 1A $cd^{4}f^{2}g$
$\He_3^2.(C_2\times D_4)$ 3O $3$ $144$ $C_3^4$ 3O 1A $cfgh^{2}$
$\He_3^2.(C_2\times D_4)$ 4A $4$ $162$ $S_3\times C_{12}$ 2E 4A $b^{3}c^{2}df^{2}h$
$\He_3^2.(C_2\times D_4)$ 4B $4$ $486$ $C_2\times C_{12}$ 2E 4B $b^{3}c^{2}d^{2}f^{2}g$
$\He_3^2.(C_2\times D_4)$ 6A $6$ $54$ $C_2\times \He_3:C_4$ 3A 2A $d^{3}h$
$\He_3^2.(C_2\times D_4)$ 6B $6$ $54$ $C_2\times \He_3:C_4$ 3A 2B $b^{2}dh$
$\He_3^2.(C_2\times D_4)$ 6C $6$ $108$ $C_3^2:D_6$ 3B 2C $ac$
$\He_3^2.(C_2\times D_4)$ 6D $6$ $108$ $C_3^2:D_6$ 3B 2D $abd^{5}f^{2}$
$\He_3^2.(C_2\times D_4)$ 6E $6$ $162$ $S_3\times C_{12}$ 3A 2E $b^{2}c^{2}d^{4}f$
$\He_3^2.(C_2\times D_4)$ 6F $6$ $162$ $C_3\times D_{12}$ 3B 2E $b^{2}f^{2}gh^{2}$
$\He_3^2.(C_2\times D_4)$ 6G $6$ $324$ $C_6\times S_3$ 3F 2F $ac^{2}de^{2}g^{2}$
$\He_3^2.(C_2\times D_4)$ 6H $6$ $324$ $C_6\times S_3$ 3G 2D $abd^{5}efgh$
$\He_3^2.(C_2\times D_4)$ 6I $6$ $324$ $C_3\times C_{12}$ 3C 2E $b^{2}d^{4}ef^{2}g^{2}h$
$\He_3^2.(C_2\times D_4)$ 6J $6$ $324$ $C_6\times S_3$ 3H 2G $ab^{3}c^{2}d^{4}e^{2}f^{2}gh^{2}$
$\He_3^2.(C_2\times D_4)$ 6K $6$ $324$ $C_6\times S_3$ 3G 2G $ab^{3}c^{2}e^{2}fg^{2}h^{2}$
$\He_3^2.(C_2\times D_4)$ 6L $6$ $324$ $C_6\times S_3$ 3I 2F $ab^{2}cd^{5}e^{2}f^{2}g$
$\He_3^2.(C_2\times D_4)$ 6M $6$ $324$ $C_6\times S_3$ 3G 2B $b^{2}cd^{3}eh$
$\He_3^2.(C_2\times D_4)$ 6N $6$ $324$ $C_6\times S_3$ 3F 2C $ad^{2}ef^{2}$
$\He_3^2.(C_2\times D_4)$ 6O $6$ $324$ $C_6\times S_3$ 3I 2A $cdg^{2}$
$\He_3^2.(C_2\times D_4)$ 6P $6$ $324$ $C_6\times S_3$ 3H 2A $cd^{3}e^{2}g^{2}h^{2}$
$\He_3^2.(C_2\times D_4)$ 6Q $6$ $324$ $C_6\times S_3$ 3F 2B $b^{2}c^{2}d^{5}efg^{2}h$
$\He_3^2.(C_2\times D_4)$ 6R $6$ $324$ $C_6\times S_3$ 3H 2D $ab^{3}c^{2}df^{2}g^{2}$
$\He_3^2.(C_2\times D_4)$ 6S $6$ $324$ $C_6\times S_3$ 3I 2C $ac^{2}d^{2}eg^{2}h^{2}$
$\He_3^2.(C_2\times D_4)$ 6T $6$ $648$ $C_3\times C_6$ 3D 2F $ab^{2}cd^{5}efgh$
$\He_3^2.(C_2\times D_4)$ 6U $6$ $648$ $C_3\times C_6$ 3L 2C $ab^{2}c^{2}d^{2}eg$
$\He_3^2.(C_2\times D_4)$ 6V $6$ $648$ $C_3\times C_6$ 3M 2D $ab^{3}c^{2}de^{2}fg^{2}h$
$\He_3^2.(C_2\times D_4)$ 6W $6$ $648$ $C_3\times C_6$ 3E 2G $ab^{3}c^{2}d^{2}e^{2}g^{2}$
$\He_3^2.(C_2\times D_4)$ 12A1 $12$ $162$ $S_3\times C_{12}$ 6E 4A $bc^{2}de^{2}g^{2}h^{2}$
$\He_3^2.(C_2\times D_4)$ 12A5 $12$ $162$ $S_3\times C_{12}$ 6E 4A $bc^{2}de^{2}g^{2}h$
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