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Elements of the group are displayed as words in the presentation $\langle a, b, c, d \mid b^{2}=c^{42}=d^{8}=[a,b]=[b,d]=1, a^{2}=d^{4}, c^{a}=cd^{4}, d^{a}=d^{7}, c^{b}=c^{13}d^{4}, d^{c}=d^{5} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P 7P
$C_{168}.C_2^3$ 1A $1$ $1$ $C_{168}.C_2^3$ 1A 1A 1A $1$
$C_{168}.C_2^3$ 2A $2$ $1$ $C_{168}.C_2^3$ 1A 2A 2A $d^{4}$
$C_{168}.C_2^3$ 2B $2$ $2$ $C_{42}:Q_{16}$ 1A 2B 2B $c^{21}$
$C_{168}.C_2^3$ 2C $2$ $4$ $C_{42}:Q_8$ 1A 2C 2C $ac^{21}$
$C_{168}.C_2^3$ 2D $2$ $14$ $C_6\times Q_{16}$ 1A 2D 2D $bc^{30}$
$C_{168}.C_2^3$ 2E $2$ $14$ $C_6\times Q_{16}$ 1A 2E 2E $bc^{15}d^{2}$
$C_{168}.C_2^3$ 2F $2$ $28$ $C_6\times Q_8$ 1A 2F 2F $abc^{15}d$
$C_{168}.C_2^3$ 3A1 $3$ $1$ $C_{168}.C_2^3$ 3A-1 1A 3A1 $c^{28}$
$C_{168}.C_2^3$ 3A-1 $3$ $1$ $C_{168}.C_2^3$ 3A1 1A 3A-1 $c^{14}$
$C_{168}.C_2^3$ 4A $4$ $2$ $C_{24}.D_{14}$ 2A 4A 4A $d^{6}$
$C_{168}.C_2^3$ 4B $4$ $2$ $C_{84}.D_4$ 2A 4B 4B $c^{21}d^{2}$
$C_{168}.C_2^3$ 4C $4$ $4$ $D_{28}:C_6$ 2A 4C 4C $a$
$C_{168}.C_2^3$ 4D $4$ $4$ $D_{28}:C_6$ 2A 4D 4D $ad$
$C_{168}.C_2^3$ 4E $4$ $4$ $D_{28}:C_6$ 2A 4E 4E $ac^{21}d$
$C_{168}.C_2^3$ 4F $4$ $14$ $D_8:C_6$ 2A 4F 4F $bc^{15}d^{4}$
$C_{168}.C_2^3$ 4G $4$ $14$ $D_8:C_6$ 2A 4G 4G $bc^{30}d^{6}$
$C_{168}.C_2^3$ 4H $4$ $28$ $D_4:C_6$ 2A 4H 4H $abc^{30}d^{4}$
$C_{168}.C_2^3$ 4I $4$ $28$ $D_4:C_6$ 2A 4I 4I $abc^{15}d^{4}$
$C_{168}.C_2^3$ 4J $4$ $28$ $D_4:C_6$ 2A 4J 4J $abc^{30}d^{5}$
$C_{168}.C_2^3$ 6A1 $6$ $1$ $C_{168}.C_2^3$ 3A1 2A 6A1 $c^{14}d^{4}$
$C_{168}.C_2^3$ 6A-1 $6$ $1$ $C_{168}.C_2^3$ 3A-1 2A 6A-1 $c^{28}d^{4}$
$C_{168}.C_2^3$ 6B1 $6$ $2$ $C_{42}:Q_{16}$ 3A-1 2B 6B1 $c^{7}$
$C_{168}.C_2^3$ 6B-1 $6$ $2$ $C_{42}:Q_{16}$ 3A1 2B 6B-1 $c^{35}$
$C_{168}.C_2^3$ 6C1 $6$ $4$ $C_{42}:Q_8$ 3A-1 2C 6C1 $ac^{7}$
$C_{168}.C_2^3$ 6C-1 $6$ $4$ $C_{42}:Q_8$ 3A1 2C 6C-1 $ac^{35}$
$C_{168}.C_2^3$ 6D1 $6$ $14$ $C_6\times Q_{16}$ 3A1 2D 6D1 $bc^{2}$
$C_{168}.C_2^3$ 6D-1 $6$ $14$ $C_6\times Q_{16}$ 3A-1 2D 6D-1 $bc^{4}$
$C_{168}.C_2^3$ 6E1 $6$ $14$ $C_6\times Q_{16}$ 3A-1 2E 6E1 $bcd^{2}$
$C_{168}.C_2^3$ 6E-1 $6$ $14$ $C_6\times Q_{16}$ 3A1 2E 6E-1 $bc^{5}d^{2}$
$C_{168}.C_2^3$ 6F1 $6$ $28$ $C_6\times Q_8$ 3A-1 2F 6F1 $abcd$
$C_{168}.C_2^3$ 6F-1 $6$ $28$ $C_6\times Q_8$ 3A1 2F 6F-1 $abc^{5}d$
$C_{168}.C_2^3$ 7A1 $7$ $2$ $Q_{16}:C_{42}$ 7A2 7A3 1A $c^{6}$
$C_{168}.C_2^3$ 7A2 $7$ $2$ $Q_{16}:C_{42}$ 7A3 7A1 1A $c^{12}$
$C_{168}.C_2^3$ 7A3 $7$ $2$ $Q_{16}:C_{42}$ 7A1 7A2 1A $c^{18}$
$C_{168}.C_2^3$ 8A $8$ $4$ $D_7\times C_{24}$ 4A 8A 8A $d^{5}$
$C_{168}.C_2^3$ 8B $8$ $4$ $D_7\times C_{24}$ 4A 8B 8B $c^{21}d^{5}$
$C_{168}.C_2^3$ 8C1 $8$ $14$ $\OD_{16}:C_6$ 4A 8C3 8C1 $bc^{30}d^{3}$
$C_{168}.C_2^3$ 8C3 $8$ $14$ $\OD_{16}:C_6$ 4A 8C1 8C3 $bc^{30}d$
$C_{168}.C_2^3$ 8D $8$ $28$ $C_2\times C_{24}$ 4A 8D 8D $bc^{15}d^{3}$
$C_{168}.C_2^3$ 12A1 $12$ $2$ $C_{24}.D_{14}$ 6A1 4A 12A1 $c^{28}d^{2}$
$C_{168}.C_2^3$ 12A-1 $12$ $2$ $C_{24}.D_{14}$ 6A-1 4A 12A-1 $c^{14}d^{6}$
$C_{168}.C_2^3$ 12B1 $12$ $2$ $C_{84}.D_4$ 6A1 4B 12B1 $c^{7}d^{6}$
$C_{168}.C_2^3$ 12B-1 $12$ $2$ $C_{84}.D_4$ 6A-1 4B 12B-1 $c^{35}d^{2}$
$C_{168}.C_2^3$ 12C1 $12$ $4$ $D_{28}:C_6$ 6A-1 4C 12C1 $ac^{14}d^{4}$
$C_{168}.C_2^3$ 12C-1 $12$ $4$ $D_{28}:C_6$ 6A1 4C 12C-1 $ac^{28}$
$C_{168}.C_2^3$ 12D1 $12$ $4$ $D_{28}:C_6$ 6A-1 4D 12D1 $ac^{14}d^{5}$
$C_{168}.C_2^3$ 12D-1 $12$ $4$ $D_{28}:C_6$ 6A1 4D 12D-1 $ac^{28}d$
$C_{168}.C_2^3$ 12E1 $12$ $4$ $D_{28}:C_6$ 6A1 4E 12E1 $ac^{7}d^{5}$
$C_{168}.C_2^3$ 12E-1 $12$ $4$ $D_{28}:C_6$ 6A-1 4E 12E-1 $ac^{35}d$
$C_{168}.C_2^3$ 12F1 $12$ $14$ $D_8:C_6$ 6A1 4F 12F1 $bc$
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