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Elements of the group are displayed as words in the presentation $\langle a, b, c \mid a^{2}=b^{4}=c^{168}=1, b^{a}=b^{3}, c^{a}=b^{2}c^{13}, c^{b}=c^{29} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P 7P
$D_{42}:\OD_{16}$ 1A $1$ $1$ $D_{42}:\OD_{16}$ 1A 1A 1A $1$
$D_{42}:\OD_{16}$ 2A $2$ $1$ $D_{42}:\OD_{16}$ 1A 2A 2A $c^{84}$
$D_{42}:\OD_{16}$ 2B $2$ $1$ $D_{42}:\OD_{16}$ 1A 2B 2B $b^{2}$
$D_{42}:\OD_{16}$ 2C $2$ $1$ $D_{42}:\OD_{16}$ 1A 2C 2C $b^{2}c^{84}$
$D_{42}:\OD_{16}$ 2D $2$ $28$ $C_2^2\times C_{12}$ 1A 2D 2D $ac^{120}$
$D_{42}:\OD_{16}$ 2E $2$ $42$ $C_2^2\times C_8$ 1A 2E 2E $abc^{4}$
$D_{42}:\OD_{16}$ 2F $2$ $42$ $C_2^2\times C_8$ 1A 2F 2F $ab^{3}c^{88}$
$D_{42}:\OD_{16}$ 3A $3$ $2$ $D_{14}:C_{24}$ 3A 1A 3A $c^{112}$
$D_{42}:\OD_{16}$ 4A1 $4$ $1$ $D_{42}:\OD_{16}$ 2A 4A-1 4A-1 $b^{2}c^{42}$
$D_{42}:\OD_{16}$ 4A-1 $4$ $1$ $D_{42}:\OD_{16}$ 2A 4A1 4A1 $b^{2}c^{126}$
$D_{42}:\OD_{16}$ 4B1 $4$ $1$ $D_{42}:\OD_{16}$ 2A 4B-1 4B-1 $c^{42}$
$D_{42}:\OD_{16}$ 4B-1 $4$ $1$ $D_{42}:\OD_{16}$ 2A 4B1 4B1 $c^{126}$
$D_{42}:\OD_{16}$ 4C $4$ $12$ $C_4\times C_{28}$ 2B 4C 4C $b^{3}$
$D_{42}:\OD_{16}$ 4D $4$ $12$ $C_4\times C_{28}$ 2C 4D 4D $b^{3}c^{14}$
$D_{42}:\OD_{16}$ 4E $4$ $28$ $C_2^2\times C_{12}$ 2A 4E 4E $ac^{30}$
$D_{42}:\OD_{16}$ 4F1 $4$ $42$ $C_2^2\times C_8$ 2A 4F-1 4F-1 $ab^{3}c^{102}$
$D_{42}:\OD_{16}$ 4F-1 $4$ $42$ $C_2^2\times C_8$ 2A 4F1 4F1 $ab^{3}c^{18}$
$D_{42}:\OD_{16}$ 6A $6$ $2$ $D_{14}:C_{24}$ 3A 2A 6A $c^{28}$
$D_{42}:\OD_{16}$ 6B $6$ $2$ $D_{14}:C_{24}$ 3A 2B 6B $b^{2}c^{56}$
$D_{42}:\OD_{16}$ 6C $6$ $2$ $D_{14}:C_{24}$ 3A 2C 6C $b^{2}c^{28}$
$D_{42}:\OD_{16}$ 6D1 $6$ $28$ $C_2^2\times C_{12}$ 3A 2D 6D1 $ac^{8}$
$D_{42}:\OD_{16}$ 6D-1 $6$ $28$ $C_2^2\times C_{12}$ 3A 2D 6D-1 $ac^{16}$
$D_{42}:\OD_{16}$ 7A1 $7$ $2$ $C_{24}:C_{28}$ 7A2 7A3 1A $c^{96}$
$D_{42}:\OD_{16}$ 7A2 $7$ $2$ $C_{24}:C_{28}$ 7A3 7A1 1A $c^{24}$
$D_{42}:\OD_{16}$ 7A3 $7$ $2$ $C_{24}:C_{28}$ 7A1 7A2 1A $c^{120}$
$D_{42}:\OD_{16}$ 8A1 $8$ $4$ $C_2\times C_{168}$ 4B1 8A-1 8A-1 $c^{21}$
$D_{42}:\OD_{16}$ 8A-1 $8$ $4$ $C_2\times C_{168}$ 4B-1 8A1 8A1 $c^{147}$
$D_{42}:\OD_{16}$ 8B1 $8$ $6$ $C_8\times D_{14}$ 4A1 8B-1 8B-1 $b^{3}c^{91}$
$D_{42}:\OD_{16}$ 8B-1 $8$ $6$ $C_8\times D_{14}$ 4A-1 8B1 8B1 $bc^{49}$
$D_{42}:\OD_{16}$ 8C1 $8$ $6$ $C_8\times D_{14}$ 4A-1 8C-1 8C-1 $b^{3}c^{105}$
$D_{42}:\OD_{16}$ 8C-1 $8$ $6$ $C_8\times D_{14}$ 4A1 8C1 8C1 $bc^{147}$
$D_{42}:\OD_{16}$ 8D1 $8$ $14$ $C_8\times D_6$ 4A1 8D3 8D-1 $ab^{2}c^{15}$
$D_{42}:\OD_{16}$ 8D-1 $8$ $14$ $C_8\times D_6$ 4A-1 8D-3 8D1 $ac^{141}$
$D_{42}:\OD_{16}$ 8D3 $8$ $14$ $C_8\times D_6$ 4A-1 8D1 8D-3 $ac^{57}$
$D_{42}:\OD_{16}$ 8D-3 $8$ $14$ $C_8\times D_6$ 4A1 8D-1 8D3 $ab^{2}c^{99}$
$D_{42}:\OD_{16}$ 8E1 $8$ $84$ $C_2\times C_8$ 4A-1 8E-1 8E-1 $ab^{3}c^{63}$
$D_{42}:\OD_{16}$ 8E-1 $8$ $84$ $C_2\times C_8$ 4A1 8E1 8E1 $abc^{105}$
$D_{42}:\OD_{16}$ 12A1 $12$ $2$ $D_{14}:C_{24}$ 6A 4A1 12A-1 $b^{2}c^{14}$
$D_{42}:\OD_{16}$ 12A-1 $12$ $2$ $D_{14}:C_{24}$ 6A 4A-1 12A1 $b^{2}c^{154}$
$D_{42}:\OD_{16}$ 12B1 $12$ $2$ $D_{14}:C_{24}$ 6A 4B1 12B-1 $c^{14}$
$D_{42}:\OD_{16}$ 12B-1 $12$ $2$ $D_{14}:C_{24}$ 6A 4B-1 12B1 $c^{154}$
$D_{42}:\OD_{16}$ 12C1 $12$ $28$ $C_2^2\times C_{12}$ 6A 4E 12C5 $ac^{2}$
$D_{42}:\OD_{16}$ 12C5 $12$ $28$ $C_2^2\times C_{12}$ 6A 4E 12C1 $ac^{10}$
$D_{42}:\OD_{16}$ 14A1 $14$ $2$ $C_{24}:C_{28}$ 7A1 14A3 2B $b^{2}c^{48}$
$D_{42}:\OD_{16}$ 14A3 $14$ $2$ $C_{24}:C_{28}$ 7A3 14A5 2B $b^{2}c^{144}$
$D_{42}:\OD_{16}$ 14A5 $14$ $2$ $C_{24}:C_{28}$ 7A2 14A1 2B $b^{2}c^{72}$
$D_{42}:\OD_{16}$ 14B1 $14$ $2$ $C_{24}:C_{28}$ 7A3 14B3 2C $b^{2}c^{60}$
$D_{42}:\OD_{16}$ 14B3 $14$ $2$ $C_{24}:C_{28}$ 7A2 14B5 2C $b^{2}c^{12}$
$D_{42}:\OD_{16}$ 14B5 $14$ $2$ $C_{24}:C_{28}$ 7A1 14B1 2C $b^{2}c^{132}$
$D_{42}:\OD_{16}$ 14C1 $14$ $2$ $C_{24}:C_{28}$ 7A3 14C3 2A $c^{60}$
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