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Elements of the group are displayed as words in the presentation $\langle a, b, c, d \mid a^{2}=b^{2}=c^{16}=d^{4}=[a,b]=[a,c]=[b,d]=[c,d]=1, d^{a}=d^{3}, c^{b}=cd^{2} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P
$D_4:(C_2\times C_{16})$ 1A $1$ $1$ $D_4:(C_2\times C_{16})$ 1A $1$
$D_4:(C_2\times C_{16})$ 2A $2$ $1$ $D_4:(C_2\times C_{16})$ 1A $d^{2}$
$D_4:(C_2\times C_{16})$ 2B $2$ $1$ $D_4:(C_2\times C_{16})$ 1A $c^{8}d^{2}$
$D_4:(C_2\times C_{16})$ 2C $2$ $1$ $D_4:(C_2\times C_{16})$ 1A $c^{8}$
$D_4:(C_2\times C_{16})$ 2D $2$ $2$ $C_4^2.C_2^3$ 1A $b$
$D_4:(C_2\times C_{16})$ 2E $2$ $2$ $C_2^3:C_{16}$ 1A $a$
$D_4:(C_2\times C_{16})$ 2F $2$ $2$ $C_4^2.C_2^3$ 1A $bc^{8}$
$D_4:(C_2\times C_{16})$ 2G $2$ $2$ $C_2^3:C_{16}$ 1A $ad$
$D_4:(C_2\times C_{16})$ 2H $2$ $2$ $C_2^3:C_{16}$ 1A $ac^{8}$
$D_4:(C_2\times C_{16})$ 2I $2$ $2$ $C_2^3:C_{16}$ 1A $ab$
$D_4:(C_2\times C_{16})$ 2J $2$ $2$ $C_2^3:C_{16}$ 1A $ac^{8}d$
$D_4:(C_2\times C_{16})$ 2K $2$ $2$ $C_2^3:C_{16}$ 1A $abd$
$D_4:(C_2\times C_{16})$ 2L $2$ $2$ $C_2^3:C_{16}$ 1A $abc^{8}$
$D_4:(C_2\times C_{16})$ 2M $2$ $2$ $C_2^3:C_{16}$ 1A $abc^{8}d$
$D_4:(C_2\times C_{16})$ 4A1 $4$ $1$ $D_4:(C_2\times C_{16})$ 2C $c^{4}d^{2}$
$D_4:(C_2\times C_{16})$ 4A-1 $4$ $1$ $D_4:(C_2\times C_{16})$ 2C $c^{12}d^{2}$
$D_4:(C_2\times C_{16})$ 4B1 $4$ $1$ $D_4:(C_2\times C_{16})$ 2C $c^{4}$
$D_4:(C_2\times C_{16})$ 4B-1 $4$ $1$ $D_4:(C_2\times C_{16})$ 2C $c^{12}$
$D_4:(C_2\times C_{16})$ 4C $4$ $2$ $C_4^2.C_8$ 2A $d$
$D_4:(C_2\times C_{16})$ 4D $4$ $2$ $C_4^2.C_8$ 2A $c^{8}d$
$D_4:(C_2\times C_{16})$ 4E $4$ $2$ $C_4^2.C_8$ 2A $bd$
$D_4:(C_2\times C_{16})$ 4F $4$ $2$ $C_4^2.C_8$ 2A $bc^{8}d$
$D_4:(C_2\times C_{16})$ 4G1 $4$ $2$ $C_4^2.C_8$ 2B $c^{4}d$
$D_4:(C_2\times C_{16})$ 4G-1 $4$ $2$ $C_4^2.C_8$ 2B $c^{12}d$
$D_4:(C_2\times C_{16})$ 4H1 $4$ $2$ $C_4^2.C_2^3$ 2C $bc^{4}$
$D_4:(C_2\times C_{16})$ 4H-1 $4$ $2$ $C_4^2.C_2^3$ 2C $bc^{12}$
$D_4:(C_2\times C_{16})$ 4I1 $4$ $2$ $C_2^3:C_{16}$ 2C $ac^{4}$
$D_4:(C_2\times C_{16})$ 4I-1 $4$ $2$ $C_2^3:C_{16}$ 2C $ac^{12}$
$D_4:(C_2\times C_{16})$ 4J1 $4$ $2$ $C_4^2.C_8$ 2B $bc^{4}d$
$D_4:(C_2\times C_{16})$ 4J-1 $4$ $2$ $C_4^2.C_8$ 2B $bc^{12}d$
$D_4:(C_2\times C_{16})$ 4K1 $4$ $2$ $C_2^3:C_{16}$ 2C $ac^{4}d$
$D_4:(C_2\times C_{16})$ 4K-1 $4$ $2$ $C_2^3:C_{16}$ 2C $ac^{12}d$
$D_4:(C_2\times C_{16})$ 4L1 $4$ $2$ $C_2^3:C_{16}$ 2C $abc^{4}$
$D_4:(C_2\times C_{16})$ 4L-1 $4$ $2$ $C_2^3:C_{16}$ 2C $abc^{12}$
$D_4:(C_2\times C_{16})$ 4M1 $4$ $2$ $C_2^3:C_{16}$ 2C $abc^{4}d$
$D_4:(C_2\times C_{16})$ 4M-1 $4$ $2$ $C_2^3:C_{16}$ 2C $abc^{12}d$
$D_4:(C_2\times C_{16})$ 8A1 $8$ $1$ $D_4:(C_2\times C_{16})$ 4B1 $c^{2}$
$D_4:(C_2\times C_{16})$ 8A-1 $8$ $1$ $D_4:(C_2\times C_{16})$ 4B-1 $c^{14}$
$D_4:(C_2\times C_{16})$ 8A3 $8$ $1$ $D_4:(C_2\times C_{16})$ 4B-1 $c^{6}$
$D_4:(C_2\times C_{16})$ 8A-3 $8$ $1$ $D_4:(C_2\times C_{16})$ 4B1 $c^{10}$
$D_4:(C_2\times C_{16})$ 8B1 $8$ $1$ $D_4:(C_2\times C_{16})$ 4B1 $c^{2}d^{2}$
$D_4:(C_2\times C_{16})$ 8B-1 $8$ $1$ $D_4:(C_2\times C_{16})$ 4B-1 $c^{14}d^{2}$
$D_4:(C_2\times C_{16})$ 8B3 $8$ $1$ $D_4:(C_2\times C_{16})$ 4B-1 $c^{6}d^{2}$
$D_4:(C_2\times C_{16})$ 8B-3 $8$ $1$ $D_4:(C_2\times C_{16})$ 4B1 $c^{10}d^{2}$
$D_4:(C_2\times C_{16})$ 8C1 $8$ $2$ $C_4^2.C_8$ 4A1 $c^{2}d$
$D_4:(C_2\times C_{16})$ 8C-1 $8$ $2$ $C_4^2.C_8$ 4A-1 $c^{14}d$
$D_4:(C_2\times C_{16})$ 8C3 $8$ $2$ $C_4^2.C_8$ 4A-1 $c^{6}d$
$D_4:(C_2\times C_{16})$ 8C-3 $8$ $2$ $C_4^2.C_8$ 4A1 $c^{10}d$
$D_4:(C_2\times C_{16})$ 8D1 $8$ $2$ $C_4^2.C_2^3$ 4B1 $bc^{2}$
$D_4:(C_2\times C_{16})$ 8D-1 $8$ $2$ $C_4^2.C_2^3$ 4B-1 $bc^{14}$
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