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Elements of the group are displayed as words in the presentation $\langle a, b, c, d, e, f, g \mid a^{2}=b^{8}=c^{6}=d^{3}=e^{3}=f^{3}=g^{3}=[a,c]=[a,f]=[d,f]=[d,g]=[e,f]=[e,g]=[f,g]=1, b^{a}=b^{3}c^{4}deg, d^{a}=d^{2}f, e^{a}=e^{2}f^{2}, g^{a}=f^{2}g^{2}, c^{b}=c^{5}de^{2}fg^{2}, d^{b}=c^{2}d^{2}f^{2}g, e^{b}=e^{2}, f^{b}=g, g^{b}=fg^{2}, d^{c}=df^{2}, e^{c}=e^{2}f^{2}g, f^{c}=f^{2}, g^{c}=g^{2}, e^{d}=ef^{2} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P
$C_3^4:(S_3\times \SD_{16})$ 1A $1$ $1$ $C_3^4:(S_3\times \SD_{16})$ 1A 1A $1$
$C_3^4:(S_3\times \SD_{16})$ 2A $2$ $27$ $F_9:C_2^2$ 1A 2A $c^{3}g^{2}$
$C_3^4:(S_3\times \SD_{16})$ 2B $2$ $27$ $F_9:C_2^2$ 1A 2B $b^{4}c^{3}f^{2}$
$C_3^4:(S_3\times \SD_{16})$ 2C $2$ $36$ $C_6.S_3^2$ 1A 2C $ac^{3}f$
$C_3^4:(S_3\times \SD_{16})$ 2D $2$ $81$ $S_3\times \SD_{16}$ 1A 2D $b^{4}c^{2}d^{2}g^{2}$
$C_3^4:(S_3\times \SD_{16})$ 2E $2$ $108$ $S_3\times D_6$ 1A 2E $ab^{2}c^{4}df$
$C_3^4:(S_3\times \SD_{16})$ 3A $3$ $8$ $C_3^3:S_3^2$ 3A 1A $fg^{2}$
$C_3^4:(S_3\times \SD_{16})$ 3B $3$ $18$ $F_9:C_6$ 3B 1A $eg^{2}$
$C_3^4:(S_3\times \SD_{16})$ 3C $3$ $24$ $C_3\wr C_2^2$ 3C 1A $c^{2}$
$C_3^4:(S_3\times \SD_{16})$ 3D $3$ $48$ $C_3^2\wr C_2$ 3D 1A $c^{2}d^{2}fg^{2}$
$C_3^4:(S_3\times \SD_{16})$ 3E $3$ $144$ $S_3\times C_3^2$ 3E 1A $d^{2}efg$
$C_3^4:(S_3\times \SD_{16})$ 4A $4$ $162$ $S_3\times C_8$ 2D 4A $b^{6}d^{2}f^{2}g$
$C_3^4:(S_3\times \SD_{16})$ 4B $4$ $324$ $C_4\times S_3$ 2D 4B $ab^{7}e^{2}f$
$C_3^4:(S_3\times \SD_{16})$ 4C $4$ $486$ $C_2\times C_8$ 2D 4C $b^{2}c^{3}d^{2}e^{2}g$
$C_3^4:(S_3\times \SD_{16})$ 4D $4$ $972$ $C_2\times C_4$ 2D 4D $ab^{7}c^{3}fg$
$C_3^4:(S_3\times \SD_{16})$ 6A $6$ $72$ $C_3^2:D_6$ 3A 2C $ac^{3}g$
$C_3^4:(S_3\times \SD_{16})$ 6B $6$ $162$ $C_3\times \SD_{16}$ 3B 2D $b^{4}c^{2}d^{2}e^{2}$
$C_3^4:(S_3\times \SD_{16})$ 6C $6$ $216$ $C_6\times S_3$ 3C 2A $cg^{2}$
$C_3^4:(S_3\times \SD_{16})$ 6D $6$ $216$ $C_6\times S_3$ 3C 2C $ab^{2}cd^{2}fg^{2}$
$C_3^4:(S_3\times \SD_{16})$ 6E $6$ $216$ $C_6\times S_3$ 3A 2B $b^{4}c^{3}$
$C_3^4:(S_3\times \SD_{16})$ 6F $6$ $216$ $C_6\times S_3$ 3B 2C $ab^{6}c^{5}de^{2}fg^{2}$
$C_3^4:(S_3\times \SD_{16})$ 6G $6$ $216$ $C_6\times S_3$ 3C 2E $ab^{2}c^{4}d^{2}e^{2}f^{2}g$
$C_3^4:(S_3\times \SD_{16})$ 6H $6$ $216$ $C_6\times S_3$ 3A 2E $ab^{6}d^{2}ef^{2}$
$C_3^4:(S_3\times \SD_{16})$ 6I $6$ $432$ $C_3\times C_6$ 3D 2E $ab^{2}c^{2}d^{2}g$
$C_3^4:(S_3\times \SD_{16})$ 6J $6$ $432$ $C_3\times C_6$ 3E 2C $ab^{4}ce^{2}fg^{2}$
$C_3^4:(S_3\times \SD_{16})$ 8A1 $8$ $162$ $S_3\times C_8$ 4A 8A1 $b^{7}cd^{2}ef^{2}g^{2}$
$C_3^4:(S_3\times \SD_{16})$ 8A-1 $8$ $162$ $S_3\times C_8$ 4A 8A-1 $bc^{3}def^{2}g^{2}$
$C_3^4:(S_3\times \SD_{16})$ 8B1 $8$ $486$ $C_2\times C_8$ 4A 8B1 $bc^{4}de$
$C_3^4:(S_3\times \SD_{16})$ 8B-1 $8$ $486$ $C_2\times C_8$ 4A 8B-1 $b^{7}def^{2}g$
$C_3^4:(S_3\times \SD_{16})$ 12A $12$ $324$ $C_{24}$ 6B 4A $b^{2}c^{2}ef$
$C_3^4:(S_3\times \SD_{16})$ 12B $12$ $648$ $C_{12}$ 6B 4B $ab^{3}c^{4}df^{2}$
$C_3^4:(S_3\times \SD_{16})$ 24A1 $24$ $324$ $C_{24}$ 12A 8A1 $b^{5}c^{5}df^{2}g^{2}$
$C_3^4:(S_3\times \SD_{16})$ 24A-1 $24$ $324$ $C_{24}$ 12A 8A-1 $b^{3}ce^{2}f^{2}$
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