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Elements of the group are displayed as words in the presentation $\langle a, b, c, d \mid a^{2}=b^{2}=c^{14}=d^{24}=[a,b]=[b,d]=1, c^{a}=cd^{18}, d^{a}=d^{7}, c^{b}=c^{13}d^{12}, d^{c}=d^{11} \rangle$ .

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 $d^{12}$
$D_{56}:D_6$ 2B $2$ $4$ $C_6:D_{28}$ 1A 2B 2B $a$
$D_{56}:D_6$ 2C $2$ $4$ $C_{42}:D_4$ 1A 2C 2C $ad^{9}$
$D_{56}:D_6$ 2D $2$ $6$ $C_{14}:D_8$ 1A 2D 2D $ac^{7}d^{3}$
$D_{56}:D_6$ 2E $2$ $12$ $C_{14}:D_4$ 1A 2E 2E $c^{7}$
$D_{56}:D_6$ 2F $2$ $14$ $C_6\times D_8$ 1A 2F 2F $b$
$D_{56}:D_6$ 2G $2$ $28$ $C_6:D_4$ 1A 2G 2G $abc^{10}d^{18}$
$D_{56}:D_6$ 2H $2$ $28$ $C_6:D_4$ 1A 2H 2H $abc^{6}d^{3}$
$D_{56}:D_6$ 2I $2$ $42$ $C_2\times D_8$ 1A 2I 2I $abc^{13}d^{5}$
$D_{56}:D_6$ 2J $2$ $84$ $C_2\times D_4$ 1A 2J 2J $bc^{13}d^{3}$
$D_{56}:D_6$ 3A $3$ $2$ $C_{24}:D_{14}$ 3A 1A 3A $d^{16}$
$D_{56}:D_6$ 4A $4$ $2$ $C_{24}.D_{14}$ 2A 4A 4A $d^{6}$
$D_{56}:D_6$ 4B $4$ $6$ $C_{28}.D_4$ 2A 4B 4B $ac^{7}d^{13}$
$D_{56}:D_6$ 4C $4$ $12$ $D_{28}:C_2$ 2A 4C 4C $c^{7}d^{13}$
$D_{56}:D_6$ 4D $4$ $14$ $D_{24}:C_2$ 2A 4D 4D $bd^{6}$
$D_{56}:D_6$ 4E $4$ $42$ $D_8:C_2$ 2A 4E 4E $abc^{13}d^{19}$
$D_{56}:D_6$ 4F $4$ $84$ $D_4:C_2$ 2A 4F 4F $bc^{5}d^{4}$
$D_{56}:D_6$ 6A $6$ $2$ $C_{24}:D_{14}$ 3A 2A 6A $d^{4}$
$D_{56}:D_6$ 6B $6$ $8$ $C_6\times D_{14}$ 3A 2B 6B $ad^{8}$
$D_{56}:D_6$ 6C $6$ $8$ $C_6\times D_{14}$ 3A 2C 6C $ad$
$D_{56}:D_6$ 6D1 $6$ $14$ $C_6\times D_8$ 3A 2F 6D1 $bd^{8}$
$D_{56}:D_6$ 6D-1 $6$ $14$ $C_6\times D_8$ 3A 2F 6D-1 $bd^{16}$
$D_{56}:D_6$ 6E $6$ $56$ $C_2^2\times C_6$ 3A 2G 6E $abc^{10}d^{2}$
$D_{56}:D_6$ 6F $6$ $56$ $C_2^2\times C_6$ 3A 2H 6F $abc^{6}d^{19}$
$D_{56}:D_6$ 7A1 $7$ $2$ $C_{56}:D_6$ 7A2 7A3 1A $c^{2}$
$D_{56}:D_6$ 7A2 $7$ $2$ $C_{56}:D_6$ 7A3 7A1 1A $c^{4}$
$D_{56}:D_6$ 7A3 $7$ $2$ $C_{56}:D_6$ 7A1 7A2 1A $c^{6}$
$D_{56}:D_6$ 8A $8$ $4$ $D_7\times C_{24}$ 4A 8A 8A $d^{21}$
$D_{56}:D_6$ 8B $8$ $12$ $C_8\times D_7$ 4A 8B 8B $ac^{7}d^{18}$
$D_{56}:D_6$ 8C1 $8$ $14$ $C_8.D_6$ 4A 8C3 8C1 $bd^{3}$
$D_{56}:D_6$ 8C3 $8$ $14$ $C_8.D_6$ 4A 8C1 8C3 $bd^{9}$
$D_{56}:D_6$ 8D $8$ $84$ $C_2\times C_8$ 4A 8D 8D $abc^{5}d^{22}$
$D_{56}:D_6$ 12A $12$ $4$ $D_7\times C_{24}$ 6A 4A 12A $d^{2}$
$D_{56}:D_6$ 12B $12$ $28$ $C_2\times C_{24}$ 6A 4D 12B $bd^{2}$
$D_{56}:D_6$ 14A1 $14$ $2$ $C_{56}:D_6$ 7A2 14A3 2A $c^{2}d^{12}$
$D_{56}:D_6$ 14A3 $14$ $2$ $C_{56}:D_6$ 7A1 14A5 2A $c^{6}d^{12}$
$D_{56}:D_6$ 14A5 $14$ $2$ $C_{56}:D_6$ 7A3 14A1 2A $c^{10}d^{12}$
$D_{56}:D_6$ 14B1 $14$ $8$ $C_6:C_{28}$ 7A1 14B3 2B $ac^{6}$
$D_{56}:D_6$ 14B3 $14$ $8$ $C_6:C_{28}$ 7A3 14B5 2B $ac^{4}$
$D_{56}:D_6$ 14B5 $14$ $8$ $C_6:C_{28}$ 7A2 14B1 2B $ac^{2}$
$D_{56}:D_6$ 14C1 $14$ $8$ $D_6\times C_{14}$ 7A1 14C3 2C $ac^{6}d^{9}$
$D_{56}:D_6$ 14C3 $14$ $8$ $D_6\times C_{14}$ 7A3 14C5 2C $ac^{4}d^{9}$
$D_{56}:D_6$ 14C5 $14$ $8$ $D_6\times C_{14}$ 7A2 14C1 2C $ac^{2}d^{9}$
$D_{56}:D_6$ 14D1 $14$ $12$ $D_4\times C_{14}$ 7A1 14D3 2D $acd^{3}$
$D_{56}:D_6$ 14D3 $14$ $12$ $D_4\times C_{14}$ 7A3 14D5 2D $ac^{3}d^{3}$
$D_{56}:D_6$ 14D5 $14$ $12$ $D_4\times C_{14}$ 7A2 14D1 2D $ac^{5}d^{3}$
$D_{56}:D_6$ 14E1 $14$ $24$ $C_2^2\times C_{14}$ 7A1 14E3 2E $c$
$D_{56}:D_6$ 14E3 $14$ $24$ $C_2^2\times C_{14}$ 7A3 14E5 2E $c^{3}$
$D_{56}:D_6$ 14E5 $14$ $24$ $C_2^2\times C_{14}$ 7A2 14E1 2E $c^{5}$
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