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Elements of the group are displayed as words in the presentation $\langle a, b, c \mid c^{168}=1, a^{2}=c^{84}, b^{4}=c^{84}, b^{a}=b^{3}, c^{a}=c^{55}, c^{b}=c^{155} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P 7P
$C_{24}.D_{28}$ 1A $1$ $1$ $C_{24}.D_{28}$ 1A 1A 1A $1$
$C_{24}.D_{28}$ 2A $2$ $1$ $C_{24}.D_{28}$ 1A 2A 2A $c^{84}$
$C_{24}.D_{28}$ 2B $2$ $2$ $C_{42}:Q_{16}$ 1A 2B 2B $b^{2}c^{42}$
$C_{24}.D_{28}$ 2C $2$ $84$ $C_2\times C_8$ 1A 2C 2C $ab^{3}c^{4}$
$C_{24}.D_{28}$ 3A $3$ $2$ $C_{42}:Q_{16}$ 3A 1A 3A $c^{112}$
$C_{24}.D_{28}$ 4A $4$ $2$ $C_{24}.C_{28}$ 2A 4A 4A $b^{2}c^{84}$
$C_{24}.D_{28}$ 4B $4$ $2$ $C_8.D_{42}$ 2A 4B 4B $c^{42}$
$C_{24}.D_{28}$ 4C $4$ $56$ $C_2\times C_{12}$ 2A 4C 4C $ac^{138}$
$C_{24}.D_{28}$ 4D $4$ $56$ $C_2\times C_{12}$ 2A 4D 4D $ac^{27}$
$C_{24}.D_{28}$ 4E $4$ $84$ $C_2\times C_8$ 2A 4E 4E $ab^{3}c^{18}$
$C_{24}.D_{28}$ 6A $6$ $2$ $C_{42}:Q_{16}$ 3A 2A 6A $c^{28}$
$C_{24}.D_{28}$ 6B1 $6$ $2$ $C_{42}:Q_{16}$ 3A 2B 6B1 $b^{2}c^{98}$
$C_{24}.D_{28}$ 6B-1 $6$ $2$ $C_{42}:Q_{16}$ 3A 2B 6B-1 $b^{2}c^{154}$
$C_{24}.D_{28}$ 7A1 $7$ $2$ $C_{24}.C_{28}$ 7A2 7A3 1A $c^{96}$
$C_{24}.D_{28}$ 7A2 $7$ $2$ $C_{24}.C_{28}$ 7A3 7A1 1A $c^{24}$
$C_{24}.D_{28}$ 7A3 $7$ $2$ $C_{24}.C_{28}$ 7A1 7A2 1A $c^{120}$
$C_{24}.D_{28}$ 8A1 $8$ $2$ $C_8.D_{42}$ 4B 8A3 8A1 $b^{2}c^{21}$
$C_{24}.D_{28}$ 8A3 $8$ $2$ $C_8.D_{42}$ 4B 8A1 8A3 $b^{2}c^{147}$
$C_{24}.D_{28}$ 8B $8$ $4$ $C_2\times C_{168}$ 4B 8B 8B $c^{21}$
$C_{24}.D_{28}$ 8C $8$ $24$ $C_{56}$ 4A 8C 8C $b^{3}c^{84}$
$C_{24}.D_{28}$ 8D $8$ $24$ $C_{56}$ 4A 8D 8D $b^{3}c^{49}$
$C_{24}.D_{28}$ 8E1 $8$ $84$ $C_2\times C_8$ 4B 8E-1 8E-1 $abc^{63}$
$C_{24}.D_{28}$ 8E-1 $8$ $84$ $C_2\times C_8$ 4B 8E1 8E1 $abc^{21}$
$C_{24}.D_{28}$ 12A $12$ $4$ $C_2\times C_{168}$ 6A 4A 12A $b^{2}c^{140}$
$C_{24}.D_{28}$ 12B $12$ $4$ $C_2\times C_{168}$ 6A 4B 12B $c^{14}$
$C_{24}.D_{28}$ 12C1 $12$ $56$ $C_2\times C_{12}$ 6A 4C 12C1 $ac^{110}$
$C_{24}.D_{28}$ 12C-1 $12$ $56$ $C_2\times C_{12}$ 6A 4C 12C-1 $ac^{82}$
$C_{24}.D_{28}$ 12D1 $12$ $56$ $C_2\times C_{12}$ 6A 4D 12D1 $ac^{167}$
$C_{24}.D_{28}$ 12D-1 $12$ $56$ $C_2\times C_{12}$ 6A 4D 12D-1 $ac^{139}$
$C_{24}.D_{28}$ 14A1 $14$ $2$ $C_{24}.C_{28}$ 7A2 14A3 2A $c^{12}$
$C_{24}.D_{28}$ 14A3 $14$ $2$ $C_{24}.C_{28}$ 7A1 14A5 2A $c^{36}$
$C_{24}.D_{28}$ 14A5 $14$ $2$ $C_{24}.C_{28}$ 7A3 14A1 2A $c^{60}$
$C_{24}.D_{28}$ 14B1 $14$ $4$ $C_2\times C_{168}$ 7A1 14B3 2B $b^{2}c^{90}$
$C_{24}.D_{28}$ 14B3 $14$ $4$ $C_2\times C_{168}$ 7A3 14B5 2B $b^{2}c^{18}$
$C_{24}.D_{28}$ 14B5 $14$ $4$ $C_2\times C_{168}$ 7A2 14B1 2B $b^{2}c^{114}$
$C_{24}.D_{28}$ 21A1 $21$ $4$ $C_2\times C_{168}$ 21A2 7A1 3A $c^{88}$
$C_{24}.D_{28}$ 21A2 $21$ $4$ $C_2\times C_{168}$ 21A4 7A2 3A $c^{8}$
$C_{24}.D_{28}$ 21A4 $21$ $4$ $C_2\times C_{168}$ 21A1 7A3 3A $c^{16}$
$C_{24}.D_{28}$ 24A1 $24$ $4$ $C_2\times C_{168}$ 12B 8B 24A1 $c^{7}$
$C_{24}.D_{28}$ 24A-1 $24$ $4$ $C_2\times C_{168}$ 12B 8B 24A-1 $c^{161}$
$C_{24}.D_{28}$ 24B1 $24$ $4$ $C_2\times C_{168}$ 12B 8A1 24B1 $b^{2}c^{91}$
$C_{24}.D_{28}$ 24B5 $24$ $4$ $C_2\times C_{168}$ 12B 8A3 24B5 $b^{2}c^{119}$
$C_{24}.D_{28}$ 28A1 $28$ $2$ $C_{24}.C_{28}$ 14A1 28A3 4A $b^{2}c^{48}$
$C_{24}.D_{28}$ 28A3 $28$ $2$ $C_{24}.C_{28}$ 14A3 28A9 4A $b^{2}c^{60}$
$C_{24}.D_{28}$ 28A5 $28$ $2$ $C_{24}.C_{28}$ 14A5 28A13 4A $b^{2}c^{72}$
$C_{24}.D_{28}$ 28A9 $28$ $2$ $C_{24}.C_{28}$ 14A5 28A1 4A $b^{2}c^{96}$
$C_{24}.D_{28}$ 28A11 $28$ $2$ $C_{24}.C_{28}$ 14A3 28A5 4A $b^{2}c^{108}$
$C_{24}.D_{28}$ 28A13 $28$ $2$ $C_{24}.C_{28}$ 14A1 28A11 4A $b^{2}c^{120}$
$C_{24}.D_{28}$ 28B1 $28$ $4$ $C_2\times C_{168}$ 14A1 28B3 4B $c^{6}$
$C_{24}.D_{28}$ 28B3 $28$ $4$ $C_2\times C_{168}$ 14A3 28B5 4B $c^{18}$
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