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Elements of the group are displayed as words in the presentation $\langle a, b \mid b^{224}=1, a^{2}=b^{112}, b^{a}=b^{127} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 7P
$C_7\times Q_{64}$ 1A $1$ $1$ $C_7\times Q_{64}$ 1A 1A $1$
$C_7\times Q_{64}$ 2A $2$ $1$ $C_7\times Q_{64}$ 1A 2A $b^{112}$
$C_7\times Q_{64}$ 4A $4$ $2$ $C_{224}$ 2A 4A $b^{56}$
$C_7\times Q_{64}$ 4B $4$ $16$ $C_{28}$ 2A 4B $ab^{112}$
$C_7\times Q_{64}$ 4C $4$ $16$ $C_{28}$ 2A 4C $ab^{49}$
$C_7\times Q_{64}$ 7A1 $7$ $1$ $C_7\times Q_{64}$ 7A2 7A3 $b^{128}$
$C_7\times Q_{64}$ 7A-1 $7$ $1$ $C_7\times Q_{64}$ 7A-2 7A-3 $b^{96}$
$C_7\times Q_{64}$ 7A2 $7$ $1$ $C_7\times Q_{64}$ 7A-3 7A-1 $b^{32}$
$C_7\times Q_{64}$ 7A-2 $7$ $1$ $C_7\times Q_{64}$ 7A3 7A1 $b^{192}$
$C_7\times Q_{64}$ 7A3 $7$ $1$ $C_7\times Q_{64}$ 7A-1 7A2 $b^{160}$
$C_7\times Q_{64}$ 7A-3 $7$ $1$ $C_7\times Q_{64}$ 7A1 7A-2 $b^{64}$
$C_7\times Q_{64}$ 8A1 $8$ $2$ $C_{224}$ 4A 8A3 $b^{28}$
$C_7\times Q_{64}$ 8A3 $8$ $2$ $C_{224}$ 4A 8A1 $b^{84}$
$C_7\times Q_{64}$ 14A1 $14$ $1$ $C_7\times Q_{64}$ 7A1 14A3 $b^{176}$
$C_7\times Q_{64}$ 14A-1 $14$ $1$ $C_7\times Q_{64}$ 7A-1 14A-3 $b^{48}$
$C_7\times Q_{64}$ 14A3 $14$ $1$ $C_7\times Q_{64}$ 7A3 14A-5 $b^{80}$
$C_7\times Q_{64}$ 14A-3 $14$ $1$ $C_7\times Q_{64}$ 7A-3 14A5 $b^{144}$
$C_7\times Q_{64}$ 14A5 $14$ $1$ $C_7\times Q_{64}$ 7A-2 14A1 $b^{208}$
$C_7\times Q_{64}$ 14A-5 $14$ $1$ $C_7\times Q_{64}$ 7A2 14A-1 $b^{16}$
$C_7\times Q_{64}$ 16A1 $16$ $2$ $C_{224}$ 8A1 16A3 $b^{14}$
$C_7\times Q_{64}$ 16A3 $16$ $2$ $C_{224}$ 8A3 16A7 $b^{42}$
$C_7\times Q_{64}$ 16A5 $16$ $2$ $C_{224}$ 8A3 16A1 $b^{70}$
$C_7\times Q_{64}$ 16A7 $16$ $2$ $C_{224}$ 8A1 16A5 $b^{98}$
$C_7\times Q_{64}$ 28A1 $28$ $2$ $C_{224}$ 14A-5 28A3 $b^{8}$
$C_7\times Q_{64}$ 28A-1 $28$ $2$ $C_{224}$ 14A5 28A-3 $b^{216}$
$C_7\times Q_{64}$ 28A3 $28$ $2$ $C_{224}$ 14A-1 28A-5 $b^{24}$
$C_7\times Q_{64}$ 28A-3 $28$ $2$ $C_{224}$ 14A1 28A5 $b^{200}$
$C_7\times Q_{64}$ 28A5 $28$ $2$ $C_{224}$ 14A3 28A1 $b^{40}$
$C_7\times Q_{64}$ 28A-5 $28$ $2$ $C_{224}$ 14A-3 28A-1 $b^{184}$
$C_7\times Q_{64}$ 28B1 $28$ $16$ $C_{28}$ 14A1 28B3 $ab^{32}$
$C_7\times Q_{64}$ 28B-1 $28$ $16$ $C_{28}$ 14A-1 28B-3 $ab^{192}$
$C_7\times Q_{64}$ 28B3 $28$ $16$ $C_{28}$ 14A3 28B-5 $ab^{96}$
$C_7\times Q_{64}$ 28B-3 $28$ $16$ $C_{28}$ 14A-3 28B5 $ab^{16}$
$C_7\times Q_{64}$ 28B5 $28$ $16$ $C_{28}$ 14A5 28B1 $ab^{160}$
$C_7\times Q_{64}$ 28B-5 $28$ $16$ $C_{28}$ 14A-5 28B-1 $ab^{8}$
$C_7\times Q_{64}$ 28C1 $28$ $16$ $C_{28}$ 14A-5 28C3 $ab$
$C_7\times Q_{64}$ 28C-1 $28$ $16$ $C_{28}$ 14A5 28C-3 $ab^{97}$
$C_7\times Q_{64}$ 28C3 $28$ $16$ $C_{28}$ 14A-1 28C-5 $ab^{17}$
$C_7\times Q_{64}$ 28C-3 $28$ $16$ $C_{28}$ 14A1 28C5 $ab^{193}$
$C_7\times Q_{64}$ 28C5 $28$ $16$ $C_{28}$ 14A3 28C1 $ab^{33}$
$C_7\times Q_{64}$ 28C-5 $28$ $16$ $C_{28}$ 14A-3 28C-1 $ab^{9}$
$C_7\times Q_{64}$ 32A1 $32$ $2$ $C_{224}$ 16A1 32A3 $b^{7}$
$C_7\times Q_{64}$ 32A3 $32$ $2$ $C_{224}$ 16A3 32A9 $b^{21}$
$C_7\times Q_{64}$ 32A5 $32$ $2$ $C_{224}$ 16A5 32A15 $b^{35}$
$C_7\times Q_{64}$ 32A7 $32$ $2$ $C_{224}$ 16A7 32A11 $b^{49}$
$C_7\times Q_{64}$ 32A9 $32$ $2$ $C_{224}$ 16A7 32A5 $b^{63}$
$C_7\times Q_{64}$ 32A11 $32$ $2$ $C_{224}$ 16A5 32A1 $b^{77}$
$C_7\times Q_{64}$ 32A13 $32$ $2$ $C_{224}$ 16A3 32A7 $b^{91}$
$C_7\times Q_{64}$ 32A15 $32$ $2$ $C_{224}$ 16A1 32A13 $b^{105}$
$C_7\times Q_{64}$ 56A1 $56$ $2$ $C_{224}$ 28A1 56A3 $b^{4}$
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