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Elements of the group are displayed as words in the presentation $\langle a, b, c, d \mid b^{2}=c^{2}=d^{168}=[a,b]=[a,c]=[b,c]=1, a^{2}=d^{84}, d^{a}=d^{85}, d^{b}=d^{29}, d^{c}=d^{55} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P 7P
$D_{56}:D_6$ 1A $1$ $1$ $D_{56}:D_6$ 1A 1A 1A $1$
$D_{56}:D_6$ 2A $2$ $1$ $D_{56}:D_6$ 1A 2A 2A $d^{84}$
$D_{56}:D_6$ 2B $2$ $2$ $D_{28}:D_6$ 1A 2B 2B $ad^{42}$
$D_{56}:D_6$ 2C $2$ $6$ $C_{14}\times \OD_{16}$ 1A 2C 2C $abd^{98}$
$D_{56}:D_6$ 2D $2$ $6$ $D_{28}:C_2^2$ 1A 2D 2D $b$
$D_{56}:D_6$ 2E $2$ $28$ $C_4\times D_6$ 1A 2E 2E $cd^{66}$
$D_{56}:D_6$ 2F $2$ $28$ $C_6.C_2^3$ 1A 2F 2F $acd^{141}$
$D_{56}:D_6$ 2G $2$ $28$ $C_6.C_2^3$ 1A 2G 2G $cd^{141}$
$D_{56}:D_6$ 2H $2$ $84$ $C_2^2\times C_4$ 1A 2H 2H $bcd^{58}$
$D_{56}:D_6$ 3A $3$ $2$ $C_{24}:D_{14}$ 3A 1A 3A $d^{112}$
$D_{56}:D_6$ 4A $4$ $2$ $C_{56}:D_6$ 2A 4A 4A $d^{42}$
$D_{56}:D_6$ 4B $4$ $2$ $D_{28}:D_6$ 2A 4B 4B $a$
$D_{56}:D_6$ 4C1 $4$ $3$ $D_{56}:C_2^2$ 2A 4C-1 4C-1 $abd^{84}$
$D_{56}:D_6$ 4C-1 $4$ $3$ $D_{56}:C_2^2$ 2A 4C1 4C1 $ab$
$D_{56}:D_6$ 4D $4$ $6$ $D_{28}:C_2^2$ 2A 4D 4D $bd^{14}$
$D_{56}:D_6$ 4E $4$ $28$ $C_4\times D_6$ 2A 4E 4E $acd^{120}$
$D_{56}:D_6$ 4F $4$ $84$ $C_2^2\times C_4$ 2A 4F 4F $bcd^{33}$
$D_{56}:D_6$ 4G $4$ $84$ $C_2^2\times C_4$ 2A 4G 4G $abcd^{9}$
$D_{56}:D_6$ 4H $4$ $84$ $C_2^2\times C_4$ 2A 4H 4H $abcd^{130}$
$D_{56}:D_6$ 6A $6$ $2$ $C_{24}:D_{14}$ 3A 2A 6A $d^{28}$
$D_{56}:D_6$ 6B $6$ $4$ $C_6\times D_{28}$ 3A 2B 6B $ad^{98}$
$D_{56}:D_6$ 6C $6$ $56$ $C_2\times C_{12}$ 3A 2E 6C $cd^{10}$
$D_{56}:D_6$ 6D $6$ $56$ $C_2^2\times C_6$ 3A 2F 6D $acd^{29}$
$D_{56}:D_6$ 6E $6$ $56$ $C_2^2\times C_6$ 3A 2G 6E $cd^{29}$
$D_{56}:D_6$ 7A1 $7$ $2$ $C_{56}:D_6$ 7A2 7A3 1A $d^{144}$
$D_{56}:D_6$ 7A2 $7$ $2$ $C_{56}:D_6$ 7A3 7A1 1A $d^{120}$
$D_{56}:D_6$ 7A3 $7$ $2$ $C_{56}:D_6$ 7A1 7A2 1A $d^{96}$
$D_{56}:D_6$ 8A $8$ $4$ $S_3\times C_{56}$ 4A 8A 8A $d^{21}$
$D_{56}:D_6$ 8B $8$ $4$ $S_3\times C_{56}$ 4A 8B 8B $ad^{21}$
$D_{56}:D_6$ 8C $8$ $12$ $C_2\times C_{56}$ 4A 8C 8C $bd^{91}$
$D_{56}:D_6$ 8D $8$ $12$ $C_2\times C_{56}$ 4A 8D 8D $abd^{91}$
$D_{56}:D_6$ 12A $12$ $4$ $D_{28}:C_6$ 6A 4B 12A $ad^{140}$
$D_{56}:D_6$ 12B $12$ $4$ $\OD_{16}\times C_{21}$ 6A 4A 12B $d^{14}$
$D_{56}:D_6$ 12C $12$ $56$ $C_2\times C_{12}$ 6A 4E 12C $acd^{148}$
$D_{56}:D_6$ 14A1 $14$ $2$ $C_{56}:D_6$ 7A3 14A3 2A $d^{132}$
$D_{56}:D_6$ 14A3 $14$ $2$ $C_{56}:D_6$ 7A2 14A5 2A $d^{60}$
$D_{56}:D_6$ 14A5 $14$ $2$ $C_{56}:D_6$ 7A1 14A1 2A $d^{156}$
$D_{56}:D_6$ 14B1 $14$ $4$ $D_6\times C_{28}$ 7A3 14B3 2B $ad^{90}$
$D_{56}:D_6$ 14B3 $14$ $4$ $D_6\times C_{28}$ 7A2 14B5 2B $ad^{18}$
$D_{56}:D_6$ 14B5 $14$ $4$ $D_6\times C_{28}$ 7A1 14B1 2B $ad^{114}$
$D_{56}:D_6$ 14C1 $14$ $6$ $C_{14}\times \OD_{16}$ 7A1 14C3 2C $abd^{2}$
$D_{56}:D_6$ 14C-1 $14$ $6$ $C_{14}\times \OD_{16}$ 7A1 14C-3 2C $abd^{26}$
$D_{56}:D_6$ 14C3 $14$ $6$ $C_{14}\times \OD_{16}$ 7A3 14C-5 2C $abd^{146}$
$D_{56}:D_6$ 14C-3 $14$ $6$ $C_{14}\times \OD_{16}$ 7A3 14C5 2C $abd^{6}$
$D_{56}:D_6$ 14C5 $14$ $6$ $C_{14}\times \OD_{16}$ 7A2 14C1 2C $abd^{10}$
$D_{56}:D_6$ 14C-5 $14$ $6$ $C_{14}\times \OD_{16}$ 7A2 14C-1 2C $abd^{18}$
$D_{56}:D_6$ 14D1 $14$ $12$ $C_2^2\times C_{28}$ 7A2 14D3 2D $bd^{24}$
$D_{56}:D_6$ 14D3 $14$ $12$ $C_2^2\times C_{28}$ 7A1 14D5 2D $bd^{16}$
$D_{56}:D_6$ 14D5 $14$ $12$ $C_2^2\times C_{28}$ 7A3 14D1 2D $bd^{8}$
$D_{56}:D_6$ 21A1 $21$ $4$ $\OD_{16}\times C_{21}$ 21A2 7A3 3A $d^{88}$
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