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Elements of the group are displayed as words in the presentation $\langle a, b, c \mid a^{2}=b^{16}=c^{34}=1, b^{a}=b^{5}c^{17}, c^{a}=b^{8}c^{33}, c^{b}=c^{21} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 17P
$D_{68}.C_8$ 1A $1$ $1$ $D_{68}.C_8$ 1A 1A $1$
$D_{68}.C_8$ 2A $2$ $1$ $D_{68}.C_8$ 1A 2A $b^{8}$
$D_{68}.C_8$ 2B $2$ $2$ $C_{34}:C_{16}$ 1A 2B $c^{17}$
$D_{68}.C_8$ 2C $2$ $68$ $C_2\times C_8$ 1A 2C $ab^{12}c^{9}$
$D_{68}.C_8$ 4A1 $4$ $1$ $D_{68}.C_8$ 2A 4A1 $b^{12}$
$D_{68}.C_8$ 4A-1 $4$ $1$ $D_{68}.C_8$ 2A 4A-1 $b^{4}$
$D_{68}.C_8$ 4B $4$ $2$ $C_{34}:C_{16}$ 2A 4B $b^{4}c^{17}$
$D_{68}.C_8$ 4C $4$ $68$ $C_2\times C_8$ 2A 4C $ab^{8}c^{19}$
$D_{68}.C_8$ 8A1 $8$ $4$ $C_8\times D_{17}$ 4A1 8A1 $ab^{2}$
$D_{68}.C_8$ 8A-1 $8$ $4$ $C_8\times D_{17}$ 4A-1 8A-1 $ab^{6}$
$D_{68}.C_8$ 8B1 $8$ $17$ $\OD_{32}:C_2$ 4A1 8B1 $b^{6}c^{3}$
$D_{68}.C_8$ 8B-1 $8$ $17$ $\OD_{32}:C_2$ 4A-1 8B-1 $b^{10}c^{3}$
$D_{68}.C_8$ 8B3 $8$ $17$ $\OD_{32}:C_2$ 4A-1 8B3 $b^{2}c^{3}$
$D_{68}.C_8$ 8B-3 $8$ $17$ $\OD_{32}:C_2$ 4A1 8B-3 $b^{14}c^{3}$
$D_{68}.C_8$ 8C1 $8$ $34$ $C_2\times C_{16}$ 4A-1 8C1 $b^{2}c^{6}$
$D_{68}.C_8$ 8C-1 $8$ $34$ $C_2\times C_{16}$ 4A1 8C-1 $b^{14}c^{6}$
$D_{68}.C_8$ 16A1 $16$ $34$ $C_2\times C_{16}$ 8C1 16A1 $b^{9}c^{8}$
$D_{68}.C_8$ 16A-1 $16$ $34$ $C_2\times C_{16}$ 8C-1 16A-1 $b^{7}c^{32}$
$D_{68}.C_8$ 16A3 $16$ $34$ $C_2\times C_{16}$ 8C-1 16A3 $b^{11}c^{32}$
$D_{68}.C_8$ 16A-3 $16$ $34$ $C_2\times C_{16}$ 8C1 16A-3 $b^{5}c^{8}$
$D_{68}.C_8$ 16A5 $16$ $34$ $C_2\times C_{16}$ 8C1 16A5 $b^{13}c^{8}$
$D_{68}.C_8$ 16A-5 $16$ $34$ $C_2\times C_{16}$ 8C-1 16A-5 $b^{3}c^{32}$
$D_{68}.C_8$ 16A7 $16$ $34$ $C_2\times C_{16}$ 8C-1 16A7 $b^{15}c^{32}$
$D_{68}.C_8$ 16A-7 $16$ $34$ $C_2\times C_{16}$ 8C1 16A-7 $bc^{8}$
$D_{68}.C_8$ 16B1 $16$ $68$ $C_{16}$ 8B1 16B1 $ab^{9}c^{16}$
$D_{68}.C_8$ 16B-1 $16$ $68$ $C_{16}$ 8B-1 16B-1 $ab^{3}c^{21}$
$D_{68}.C_8$ 16B3 $16$ $68$ $C_{16}$ 8B3 16B3 $ab^{15}c^{21}$
$D_{68}.C_8$ 16B-3 $16$ $68$ $C_{16}$ 8B-3 16B-3 $ab^{13}c^{16}$
$D_{68}.C_8$ 17A1 $17$ $4$ $\OD_{16}\times C_{17}$ 17A2 1A $c^{2}$
$D_{68}.C_8$ 17A2 $17$ $4$ $\OD_{16}\times C_{17}$ 17A1 1A $c^{4}$
$D_{68}.C_8$ 17A3 $17$ $4$ $\OD_{16}\times C_{17}$ 17A6 1A $c^{6}$
$D_{68}.C_8$ 17A6 $17$ $4$ $\OD_{16}\times C_{17}$ 17A3 1A $c^{12}$
$D_{68}.C_8$ 34A1 $34$ $4$ $\OD_{16}\times C_{17}$ 17A2 2A $b^{8}c^{2}$
$D_{68}.C_8$ 34A3 $34$ $4$ $\OD_{16}\times C_{17}$ 17A6 2A $b^{8}c^{6}$
$D_{68}.C_8$ 34A7 $34$ $4$ $\OD_{16}\times C_{17}$ 17A3 2A $b^{8}c^{14}$
$D_{68}.C_8$ 34A9 $34$ $4$ $\OD_{16}\times C_{17}$ 17A1 2A $b^{8}c^{18}$
$D_{68}.C_8$ 34B1 $34$ $8$ $C_2\times C_{68}$ 17A1 2B $c$
$D_{68}.C_8$ 34B3 $34$ $8$ $C_2\times C_{68}$ 17A3 2B $c^{3}$
$D_{68}.C_8$ 34B7 $34$ $8$ $C_2\times C_{68}$ 17A6 2B $c^{7}$
$D_{68}.C_8$ 34B9 $34$ $8$ $C_2\times C_{68}$ 17A2 2B $c^{9}$
$D_{68}.C_8$ 68A1 $68$ $4$ $\OD_{16}\times C_{17}$ 34A1 4A1 $b^{12}c^{4}$
$D_{68}.C_8$ 68A-1 $68$ $4$ $\OD_{16}\times C_{17}$ 34A1 4A-1 $b^{4}c^{30}$
$D_{68}.C_8$ 68A3 $68$ $4$ $\OD_{16}\times C_{17}$ 34A3 4A-1 $b^{4}c^{12}$
$D_{68}.C_8$ 68A-3 $68$ $4$ $\OD_{16}\times C_{17}$ 34A3 4A1 $b^{12}c^{22}$
$D_{68}.C_8$ 68A7 $68$ $4$ $\OD_{16}\times C_{17}$ 34A7 4A-1 $b^{4}c^{28}$
$D_{68}.C_8$ 68A-7 $68$ $4$ $\OD_{16}\times C_{17}$ 34A7 4A1 $b^{12}c^{6}$
$D_{68}.C_8$ 68A9 $68$ $4$ $\OD_{16}\times C_{17}$ 34A9 4A1 $b^{12}c^{2}$
$D_{68}.C_8$ 68A-9 $68$ $4$ $\OD_{16}\times C_{17}$ 34A9 4A-1 $b^{4}c^{32}$
$D_{68}.C_8$ 68B1 $68$ $8$ $C_2\times C_{68}$ 34A1 4B $b^{4}c$
$D_{68}.C_8$ 68B3 $68$ $8$ $C_2\times C_{68}$ 34A3 4B $b^{4}c^{3}$
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