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Elements of the group are displayed as words in the presentation $\langle a, b, c, d \mid a^{2}=b^{2}=c^{4}=d^{84}=[a,b]=[a,d]=1, c^{a}=c^{3}d^{21}, c^{b}=cd^{21}, d^{b}=d^{83}, d^{c}=d^{29} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P 7P
$D_{84}.D_4$ 1A $1$ $1$ $D_{84}.D_4$ 1A 1A 1A $1$
$D_{84}.D_4$ 2A $2$ $1$ $D_{84}.D_4$ 1A 2A 2A $d^{42}$
$D_{84}.D_4$ 2B $2$ $2$ $C_{12}.D_{28}$ 1A 2B 2B $c^{2}$
$D_{84}.D_4$ 2C $2$ $4$ $C_2\times D_{84}$ 1A 2C 2C $a$
$D_{84}.D_4$ 2D $2$ $84$ $C_2\times D_4$ 1A 2D 2D $abd^{36}$
$D_{84}.D_4$ 2E $2$ $84$ $C_2\times D_4$ 1A 2E 2E $bd^{44}$
$D_{84}.D_4$ 2F $2$ $84$ $C_2\times D_4$ 1A 2F 2F $abc^{2}d^{77}$
$D_{84}.D_4$ 3A $3$ $2$ $C_{84}.D_4$ 3A 1A 3A $d^{56}$
$D_{84}.D_4$ 4A $4$ $2$ $C_{84}.D_4$ 2A 4A 4A $d^{63}$
$D_{84}.D_4$ 4B $4$ $2$ $D_{84}:C_4$ 2A 4B 4B $c^{2}d^{21}$
$D_{84}.D_4$ 4C $4$ $4$ $D_{84}:C_2$ 2A 4C 4C $ad^{21}$
$D_{84}.D_4$ 4D1 $4$ $12$ $C_4\times C_{28}$ 2B 4D-1 4D-1 $c^{3}$
$D_{84}.D_4$ 4D-1 $4$ $12$ $C_4\times C_{28}$ 2B 4D1 4D1 $c$
$D_{84}.D_4$ 4E $4$ $56$ $C_2\times C_{12}$ 2A 4E 4E $abc$
$D_{84}.D_4$ 4F $4$ $84$ $D_4:C_2$ 2A 4F 4F $bc^{2}d^{45}$
$D_{84}.D_4$ 6A $6$ $2$ $C_{84}.D_4$ 3A 2A 6A $d^{14}$
$D_{84}.D_4$ 6B $6$ $4$ $C_{42}:Q_8$ 3A 2B 6B $c^{2}d^{28}$
$D_{84}.D_4$ 6C $6$ $8$ $C_2\times C_{84}$ 3A 2C 6C $ad^{28}$
$D_{84}.D_4$ 7A1 $7$ $2$ $C_{84}.D_4$ 7A2 7A3 1A $d^{48}$
$D_{84}.D_4$ 7A2 $7$ $2$ $C_{84}.D_4$ 7A3 7A1 1A $d^{12}$
$D_{84}.D_4$ 7A3 $7$ $2$ $C_{84}.D_4$ 7A1 7A2 1A $d^{60}$
$D_{84}.D_4$ 8A $8$ $24$ $C_{56}$ 4A 8A 8A $acd^{63}$
$D_{84}.D_4$ 8B $8$ $56$ $C_{24}$ 4B 8B 8B $bc^{3}d^{75}$
$D_{84}.D_4$ 12A $12$ $4$ $D_4:C_{42}$ 6A 4A 12A $d^{7}$
$D_{84}.D_4$ 12B $12$ $4$ $C_{21}:\OD_{16}$ 6A 4B 12B $c^{2}d^{7}$
$D_{84}.D_4$ 12C $12$ $8$ $C_2\times C_{84}$ 6A 4C 12C $ad^{7}$
$D_{84}.D_4$ 12D1 $12$ $56$ $C_2\times C_{12}$ 6A 4E 12D1 $abcd^{70}$
$D_{84}.D_4$ 12D-1 $12$ $56$ $C_2\times C_{12}$ 6A 4E 12D-1 $abcd^{56}$
$D_{84}.D_4$ 14A1 $14$ $2$ $C_{84}.D_4$ 7A2 14A3 2A $d^{6}$
$D_{84}.D_4$ 14A3 $14$ $2$ $C_{84}.D_4$ 7A1 14A5 2A $d^{18}$
$D_{84}.D_4$ 14A5 $14$ $2$ $C_{84}.D_4$ 7A3 14A1 2A $d^{30}$
$D_{84}.D_4$ 14B1 $14$ $4$ $C_{12}:C_{28}$ 7A1 14B3 2B $c^{2}d^{24}$
$D_{84}.D_4$ 14B3 $14$ $4$ $C_{12}:C_{28}$ 7A3 14B5 2B $c^{2}d^{72}$
$D_{84}.D_4$ 14B5 $14$ $4$ $C_{12}:C_{28}$ 7A2 14B1 2B $c^{2}d^{36}$
$D_{84}.D_4$ 14C1 $14$ $8$ $C_2\times C_{84}$ 7A3 14C3 2C $ad^{12}$
$D_{84}.D_4$ 14C3 $14$ $8$ $C_2\times C_{84}$ 7A2 14C5 2C $ad^{36}$
$D_{84}.D_4$ 14C5 $14$ $8$ $C_2\times C_{84}$ 7A1 14C1 2C $ad^{60}$
$D_{84}.D_4$ 21A1 $21$ $4$ $D_4:C_{42}$ 21A2 7A3 3A $d^{8}$
$D_{84}.D_4$ 21A2 $21$ $4$ $D_4:C_{42}$ 21A4 7A1 3A $d^{16}$
$D_{84}.D_4$ 21A4 $21$ $4$ $D_4:C_{42}$ 21A1 7A2 3A $d^{32}$
$D_{84}.D_4$ 24A1 $24$ $56$ $C_{24}$ 12B 8B 24A1 $bcd^{82}$
$D_{84}.D_4$ 24A-1 $24$ $56$ $C_{24}$ 12B 8B 24A-1 $bc^{3}d^{5}$
$D_{84}.D_4$ 28A1 $28$ $2$ $C_{84}.D_4$ 14A1 28A3 4A $d^{45}$
$D_{84}.D_4$ 28A3 $28$ $2$ $C_{84}.D_4$ 14A3 28A9 4A $d^{51}$
$D_{84}.D_4$ 28A5 $28$ $2$ $C_{84}.D_4$ 14A5 28A13 4A $d^{57}$
$D_{84}.D_4$ 28A9 $28$ $2$ $C_{84}.D_4$ 14A5 28A1 4A $d^{69}$
$D_{84}.D_4$ 28A11 $28$ $2$ $C_{84}.D_4$ 14A3 28A5 4A $d^{75}$
$D_{84}.D_4$ 28A13 $28$ $2$ $C_{84}.D_4$ 14A1 28A11 4A $d^{81}$
$D_{84}.D_4$ 28B1 $28$ $4$ $C_{12}:C_{28}$ 14A1 28B3 4B $c^{2}d^{3}$
$D_{84}.D_4$ 28B3 $28$ $4$ $C_{12}:C_{28}$ 14A3 28B5 4B $c^{2}d^{9}$
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