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Elements of the group are displayed as words in the presentation $\langle a, b, c, d, e, f \mid b^{4}=c^{2}=d^{4}=e^{4}=f^{4}=[b,c]=[c,e]=[e,f]=1, a^{2}=b^{3}ce^{3}f^{3}, b^{a}=b^{3}cd^{2}e^{2}f, c^{a}=ce^{2}f^{2}, d^{a}=bcd^{3}e^{3}f, e^{a}=ce, f^{a}=b^{2}cd^{2}e^{2}f^{2}, d^{b}=d^{3}e^{3}f, e^{b}=e^{3}f^{2}, f^{b}=ce^{3}f, d^{c}=df^{2}, f^{c}=e^{2}f, e^{d}=ef^{2}, f^{d}=ef \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P
$C_2^4.C_2\wr C_4$ 1A $1$ $1$ $C_2^4.C_2\wr C_4$ 1A $1$
$C_2^4.C_2\wr C_4$ 2A $2$ $1$ $C_2^4.C_2\wr C_4$ 1A $e^{2}$
$C_2^4.C_2\wr C_4$ 2B $2$ $2$ $C_2^4.C_2^4.C_2$ 1A $f^{2}$
$C_2^4.C_2\wr C_4$ 2C $2$ $4$ $C_2^2.D_4^2$ 1A $b^{2}e^{2}$
$C_2^4.C_2\wr C_4$ 2D $2$ $4$ $C_2^2.D_4^2$ 1A $ce^{2}$
$C_2^4.C_2\wr C_4$ 2E $2$ $4$ $C_2^2.D_4^2$ 1A $b^{2}$
$C_2^4.C_2\wr C_4$ 2F $2$ $16$ $C_2^3.D_4$ 1A $b^{2}f$
$C_2^4.C_2\wr C_4$ 2G $2$ $32$ $C_2^5$ 1A $ade$
$C_2^4.C_2\wr C_4$ 2H $2$ $32$ $C_2^5$ 1A $adef^{2}$
$C_2^4.C_2\wr C_4$ 4A $4$ $8$ $(C_2^3\times C_4):C_4$ 2A $e$
$C_2^4.C_2\wr C_4$ 4B $4$ $8$ $C_2^4.C_8$ 2A $b^{2}ce^{3}f^{2}$
$C_2^4.C_2\wr C_4$ 4C $4$ $8$ $C_4^2:D_4$ 2A $b^{2}cd^{2}e^{2}f^{3}$
$C_2^4.C_2\wr C_4$ 4D $4$ $8$ $C_4^2:D_4$ 2A $b^{2}cd^{2}ef^{3}$
$C_2^4.C_2\wr C_4$ 4E $4$ $16$ $C_2^2\times C_4^2$ 2A $d^{2}f$
$C_2^4.C_2\wr C_4$ 4F $4$ $16$ $C_4^2:C_2^2$ 2A $b^{2}ef$
$C_2^4.C_2\wr C_4$ 4G1 $4$ $16$ $C_2^2\times C_4^2$ 2D $ad$
$C_2^4.C_2\wr C_4$ 4G-1 $4$ $16$ $C_2^2\times C_4^2$ 2D $ade^{2}$
$C_2^4.C_2\wr C_4$ 4H1 $4$ $16$ $C_2^2\times C_4^2$ 2D $adf^{2}$
$C_2^4.C_2\wr C_4$ 4H-1 $4$ $16$ $C_2^2\times C_4^2$ 2D $ade^{2}f^{2}$
$C_2^4.C_2\wr C_4$ 4I $4$ $32$ $C_4\times Q_8$ 2B $f$
$C_2^4.C_2\wr C_4$ 4J $4$ $32$ $C_2^3\times C_4$ 2E $b$
$C_2^4.C_2\wr C_4$ 4K $4$ $32$ $C_2^3\times C_4$ 2E $bf^{2}$
$C_2^4.C_2\wr C_4$ 4L1 $4$ $64$ $C_4^2$ 2F $b^{3}de^{2}f$
$C_2^4.C_2\wr C_4$ 4L-1 $4$ $64$ $C_4^2$ 2F $bcdf^{2}$
$C_2^4.C_2\wr C_4$ 8A1 $8$ $32$ $\OD_{32}$ 4B $bf$
$C_2^4.C_2\wr C_4$ 8A-1 $8$ $32$ $\OD_{32}$ 4B $b^{3}ce^{3}f$
$C_2^4.C_2\wr C_4$ 8B $8$ $64$ $C_2\times C_8$ 4C $ab^{2}de^{2}f$
$C_2^4.C_2\wr C_4$ 8C $8$ $64$ $C_2\times C_8$ 4D $abcde^{2}f$
$C_2^4.C_2\wr C_4$ 8D1 $8$ $64$ $C_2\times C_8$ 4F $b^{3}d^{3}ef^{3}$
$C_2^4.C_2\wr C_4$ 8D-1 $8$ $64$ $C_2\times C_8$ 4F $bcd^{3}e^{2}f^{2}$
$C_2^4.C_2\wr C_4$ 16A1 $16$ $64$ $C_{16}$ 8A1 $ab^{3}ef$
$C_2^4.C_2\wr C_4$ 16A-1 $16$ $64$ $C_{16}$ 8A-1 $ab^{2}e^{3}f^{2}$
$C_2^4.C_2\wr C_4$ 16A3 $16$ $64$ $C_{16}$ 8A-1 $ace^{2}$
$C_2^4.C_2\wr C_4$ 16A-3 $16$ $64$ $C_{16}$ 8A1 $abcf$
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