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Elements of the group are displayed as words in the presentation $\langle a, b, c, d \mid a^{2}=b^{2}=d^{12}=[a,b]=[a,d]=[b,d]=1, c^{28}=d^{6}, c^{a}=c^{15}, c^{b}=c^{13}, d^{c}=d^{11} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P 7P
$C_{84}.C_2^4$ 1A $1$ $1$ $C_{84}.C_2^4$ 1A 1A 1A $1$
$C_{84}.C_2^4$ 2A $2$ $1$ $C_{84}.C_2^4$ 1A 2A 2A $d^{6}$
$C_{84}.C_2^4$ 2B $2$ $2$ $D_{12}:D_{14}$ 1A 2B 2B $c^{14}d^{3}$
$C_{84}.C_2^4$ 2C $2$ $4$ $C_{12}\times D_{14}$ 1A 2C 2C $ad^{6}$
$C_{84}.C_2^4$ 2D $2$ $12$ $C_2^2.D_{14}$ 1A 2D 2D $ac^{21}d^{6}$
$C_{84}.C_2^4$ 2E $2$ $12$ $C_2^2.D_{14}$ 1A 2E 2E $ac^{21}d^{7}$
$C_{84}.C_2^4$ 2F $2$ $14$ $C_6.C_2^4$ 1A 2F 2F $b$
$C_{84}.C_2^4$ 2G $2$ $14$ $C_6:\OD_{16}$ 1A 2G 2G $bc^{2}d^{9}$
$C_{84}.C_2^4$ 2H $2$ $28$ $C_2^2\times C_{12}$ 1A 2H 2H $ab$
$C_{84}.C_2^4$ 3A $3$ $2$ $C_{84}.C_2^3$ 3A 1A 3A $d^{8}$
$C_{84}.C_2^4$ 4A $4$ $2$ $C_{84}.C_2^3$ 2A 4A 4A $c^{14}$
$C_{84}.C_2^4$ 4B $4$ $2$ $C_{84}.C_2^3$ 2A 4B 4B $d^{3}$
$C_{84}.C_2^4$ 4C $4$ $4$ $C_{12}\times D_{14}$ 2A 4C 4C $ad^{3}$
$C_{84}.C_2^4$ 4D1 $4$ $7$ $C_{12}.C_2^4$ 2A 4D-1 4D-1 $bd^{3}$
$C_{84}.C_2^4$ 4D-1 $4$ $7$ $C_{12}.C_2^4$ 2A 4D1 4D1 $bd^{9}$
$C_{84}.C_2^4$ 4E $4$ $14$ $D_{12}:C_2^2$ 2A 4E 4E $bc^{2}d^{6}$
$C_{84}.C_2^4$ 4F $4$ $28$ $C_2^2\times C_{12}$ 2A 4F 4F $abd^{3}$
$C_{84}.C_2^4$ 4G $4$ $84$ $C_2^2\times C_4$ 2A 4G 4G $abc^{17}d^{10}$
$C_{84}.C_2^4$ 4H $4$ $84$ $C_2^2\times C_4$ 2A 4H 4H $abc^{17}d^{9}$
$C_{84}.C_2^4$ 6A $6$ $2$ $C_{84}.C_2^3$ 3A 2A 6A $d^{2}$
$C_{84}.C_2^4$ 6B $6$ $4$ $C_{12}\times D_{14}$ 3A 2B 6B $c^{14}d^{7}$
$C_{84}.C_2^4$ 6C1 $6$ $4$ $C_{12}\times D_{14}$ 3A 2C 6C1 $ad^{10}$
$C_{84}.C_2^4$ 6C-1 $6$ $4$ $C_{12}\times D_{14}$ 3A 2C 6C-1 $ad^{2}$
$C_{84}.C_2^4$ 6D1 $6$ $14$ $C_6.C_2^4$ 3A 2F 6D1 $bd^{4}$
$C_{84}.C_2^4$ 6D-1 $6$ $14$ $C_6.C_2^4$ 3A 2F 6D-1 $bd^{8}$
$C_{84}.C_2^4$ 6E $6$ $28$ $C_2^2\times C_{12}$ 3A 2G 6E $bc^{2}d$
$C_{84}.C_2^4$ 6F1 $6$ $28$ $C_2^2\times C_{12}$ 3A 2H 6F1 $abd^{4}$
$C_{84}.C_2^4$ 6F-1 $6$ $28$ $C_2^2\times C_{12}$ 3A 2H 6F-1 $abd^{2}$
$C_{84}.C_2^4$ 7A1 $7$ $2$ $C_{84}.D_4$ 7A2 7A3 1A $c^{16}$
$C_{84}.C_2^4$ 7A2 $7$ $2$ $C_{84}.D_4$ 7A3 7A1 1A $c^{4}d^{6}$
$C_{84}.C_2^4$ 7A3 $7$ $2$ $C_{84}.D_4$ 7A1 7A2 1A $c^{20}d^{6}$
$C_{84}.C_2^4$ 8A $8$ $12$ $C_8\times D_7$ 4A 8A 8A $c^{7}$
$C_{84}.C_2^4$ 8B $8$ $12$ $C_8\times D_7$ 4A 8B 8B $c^{7}d$
$C_{84}.C_2^4$ 8C $8$ $84$ $C_2\times C_8$ 4A 8C 8C $bc^{17}d^{2}$
$C_{84}.C_2^4$ 8D $8$ $84$ $C_2\times C_8$ 4A 8D 8D $bc^{13}d^{5}$
$C_{84}.C_2^4$ 12A1 $12$ $2$ $C_{84}.C_2^3$ 6A 4B 12A5 $d$
$C_{84}.C_2^4$ 12A5 $12$ $2$ $C_{84}.C_2^3$ 6A 4B 12A1 $d^{5}$
$C_{84}.C_2^4$ 12B $12$ $4$ $C_{12}\times D_{14}$ 6A 4A 12B $c^{14}d^{4}$
$C_{84}.C_2^4$ 12C1 $12$ $4$ $C_{12}\times D_{14}$ 6A 4C 12C1 $ad$
$C_{84}.C_2^4$ 12C-1 $12$ $4$ $C_{12}\times D_{14}$ 6A 4C 12C-1 $ad^{11}$
$C_{84}.C_2^4$ 12D1 $12$ $14$ $C_6.C_2^4$ 6A 4D1 12D-1 $bd$
$C_{84}.C_2^4$ 12D-1 $12$ $14$ $C_6.C_2^4$ 6A 4D-1 12D1 $bc^{4}d$
$C_{84}.C_2^4$ 12E $12$ $28$ $C_2^2\times C_{12}$ 6A 4E 12E $bc^{2}d^{4}$
$C_{84}.C_2^4$ 12F1 $12$ $28$ $C_2^2\times C_{12}$ 6A 4F 12F1 $abd$
$C_{84}.C_2^4$ 12F-1 $12$ $28$ $C_2^2\times C_{12}$ 6A 4F 12F-1 $abd^{5}$
$C_{84}.C_2^4$ 14A1 $14$ $2$ $C_{84}.D_4$ 7A3 14A3 2A $c^{4}$
$C_{84}.C_2^4$ 14A3 $14$ $2$ $C_{84}.D_4$ 7A2 14A5 2A $c^{12}$
$C_{84}.C_2^4$ 14A5 $14$ $2$ $C_{84}.D_4$ 7A1 14A1 2A $c^{20}$
$C_{84}.C_2^4$ 14B1 $14$ $4$ $C_{14}\times D_{12}$ 7A1 14B3 2B $c^{6}d^{3}$
$C_{84}.C_2^4$ 14B3 $14$ $4$ $C_{14}\times D_{12}$ 7A3 14B5 2B $c^{18}d^{9}$
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