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Elements of the group are displayed as words in the presentation $\langle a, b \mid b^{168}=1, a^{2}=b^{84}, b^{a}=b^{127} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P 7P
$Q_{16}\times C_{21}$ 1A $1$ $1$ $Q_{16}\times C_{21}$ 1A 1A 1A $1$
$Q_{16}\times C_{21}$ 2A $2$ $1$ $Q_{16}\times C_{21}$ 1A 2A 2A $b^{84}$
$Q_{16}\times C_{21}$ 3A1 $3$ $1$ $Q_{16}\times C_{21}$ 3A-1 1A 3A1 $b^{56}$
$Q_{16}\times C_{21}$ 3A-1 $3$ $1$ $Q_{16}\times C_{21}$ 3A1 1A 3A-1 $b^{112}$
$Q_{16}\times C_{21}$ 4A $4$ $2$ $C_{168}$ 2A 4A 4A $b^{42}$
$Q_{16}\times C_{21}$ 4B $4$ $4$ $C_{84}$ 2A 4B 4B $a$
$Q_{16}\times C_{21}$ 4C $4$ $4$ $C_{84}$ 2A 4C 4C $ab^{105}$
$Q_{16}\times C_{21}$ 6A1 $6$ $1$ $Q_{16}\times C_{21}$ 3A1 2A 6A1 $b^{28}$
$Q_{16}\times C_{21}$ 6A-1 $6$ $1$ $Q_{16}\times C_{21}$ 3A-1 2A 6A-1 $b^{140}$
$Q_{16}\times C_{21}$ 7A1 $7$ $1$ $Q_{16}\times C_{21}$ 7A2 7A3 1A $b^{96}$
$Q_{16}\times C_{21}$ 7A-1 $7$ $1$ $Q_{16}\times C_{21}$ 7A-2 7A-3 1A $b^{72}$
$Q_{16}\times C_{21}$ 7A2 $7$ $1$ $Q_{16}\times C_{21}$ 7A-3 7A-1 1A $b^{24}$
$Q_{16}\times C_{21}$ 7A-2 $7$ $1$ $Q_{16}\times C_{21}$ 7A3 7A1 1A $b^{144}$
$Q_{16}\times C_{21}$ 7A3 $7$ $1$ $Q_{16}\times C_{21}$ 7A-1 7A2 1A $b^{120}$
$Q_{16}\times C_{21}$ 7A-3 $7$ $1$ $Q_{16}\times C_{21}$ 7A1 7A-2 1A $b^{48}$
$Q_{16}\times C_{21}$ 8A1 $8$ $2$ $C_{168}$ 4A 8A3 8A1 $b^{21}$
$Q_{16}\times C_{21}$ 8A3 $8$ $2$ $C_{168}$ 4A 8A1 8A3 $b^{63}$
$Q_{16}\times C_{21}$ 12A1 $12$ $2$ $C_{168}$ 6A1 4A 12A1 $b^{14}$
$Q_{16}\times C_{21}$ 12A-1 $12$ $2$ $C_{168}$ 6A-1 4A 12A-1 $b^{154}$
$Q_{16}\times C_{21}$ 12B1 $12$ $4$ $C_{84}$ 6A1 4B 12B1 $ab^{140}$
$Q_{16}\times C_{21}$ 12B-1 $12$ $4$ $C_{84}$ 6A-1 4B 12B-1 $ab^{112}$
$Q_{16}\times C_{21}$ 12C1 $12$ $4$ $C_{84}$ 6A-1 4C 12C1 $ab^{133}$
$Q_{16}\times C_{21}$ 12C-1 $12$ $4$ $C_{84}$ 6A1 4C 12C-1 $ab^{161}$
$Q_{16}\times C_{21}$ 14A1 $14$ $1$ $Q_{16}\times C_{21}$ 7A1 14A3 2A $b^{132}$
$Q_{16}\times C_{21}$ 14A-1 $14$ $1$ $Q_{16}\times C_{21}$ 7A-1 14A-3 2A $b^{36}$
$Q_{16}\times C_{21}$ 14A3 $14$ $1$ $Q_{16}\times C_{21}$ 7A3 14A-5 2A $b^{60}$
$Q_{16}\times C_{21}$ 14A-3 $14$ $1$ $Q_{16}\times C_{21}$ 7A-3 14A5 2A $b^{108}$
$Q_{16}\times C_{21}$ 14A5 $14$ $1$ $Q_{16}\times C_{21}$ 7A-2 14A1 2A $b^{156}$
$Q_{16}\times C_{21}$ 14A-5 $14$ $1$ $Q_{16}\times C_{21}$ 7A2 14A-1 2A $b^{12}$
$Q_{16}\times C_{21}$ 21A1 $21$ $1$ $Q_{16}\times C_{21}$ 21A2 7A1 3A1 $b^{32}$
$Q_{16}\times C_{21}$ 21A-1 $21$ $1$ $Q_{16}\times C_{21}$ 21A-2 7A-1 3A-1 $b^{136}$
$Q_{16}\times C_{21}$ 21A2 $21$ $1$ $Q_{16}\times C_{21}$ 21A4 7A2 3A-1 $b^{64}$
$Q_{16}\times C_{21}$ 21A-2 $21$ $1$ $Q_{16}\times C_{21}$ 21A-4 7A-2 3A1 $b^{104}$
$Q_{16}\times C_{21}$ 21A4 $21$ $1$ $Q_{16}\times C_{21}$ 21A8 7A-3 3A1 $b^{128}$
$Q_{16}\times C_{21}$ 21A-4 $21$ $1$ $Q_{16}\times C_{21}$ 21A-8 7A3 3A-1 $b^{40}$
$Q_{16}\times C_{21}$ 21A5 $21$ $1$ $Q_{16}\times C_{21}$ 21A10 7A-2 3A-1 $b^{160}$
$Q_{16}\times C_{21}$ 21A-5 $21$ $1$ $Q_{16}\times C_{21}$ 21A-10 7A2 3A1 $b^{8}$
$Q_{16}\times C_{21}$ 21A8 $21$ $1$ $Q_{16}\times C_{21}$ 21A-5 7A1 3A-1 $b^{88}$
$Q_{16}\times C_{21}$ 21A-8 $21$ $1$ $Q_{16}\times C_{21}$ 21A5 7A-1 3A1 $b^{80}$
$Q_{16}\times C_{21}$ 21A10 $21$ $1$ $Q_{16}\times C_{21}$ 21A-1 7A3 3A1 $b^{152}$
$Q_{16}\times C_{21}$ 21A-10 $21$ $1$ $Q_{16}\times C_{21}$ 21A1 7A-3 3A-1 $b^{16}$
$Q_{16}\times C_{21}$ 24A1 $24$ $2$ $C_{168}$ 12A1 8A1 24A1 $b^{7}$
$Q_{16}\times C_{21}$ 24A-1 $24$ $2$ $C_{168}$ 12A-1 8A1 24A-1 $b^{161}$
$Q_{16}\times C_{21}$ 24A5 $24$ $2$ $C_{168}$ 12A-1 8A3 24A5 $b^{35}$
$Q_{16}\times C_{21}$ 24A-5 $24$ $2$ $C_{168}$ 12A1 8A3 24A-5 $b^{133}$
$Q_{16}\times C_{21}$ 28A1 $28$ $2$ $C_{168}$ 14A-5 28A3 4A $b^{6}$
$Q_{16}\times C_{21}$ 28A-1 $28$ $2$ $C_{168}$ 14A5 28A-3 4A $b^{162}$
$Q_{16}\times C_{21}$ 28A3 $28$ $2$ $C_{168}$ 14A-1 28A-5 4A $b^{18}$
$Q_{16}\times C_{21}$ 28A-3 $28$ $2$ $C_{168}$ 14A1 28A5 4A $b^{150}$
$Q_{16}\times C_{21}$ 28A5 $28$ $2$ $C_{168}$ 14A3 28A1 4A $b^{30}$
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