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Elements of the group are displayed as words in the presentation $\langle a, b, c \mid b^{248}=c^{2}=[a,c]=[b,c]=1, a^{4}=b^{124}, b^{a}=b^{123}c \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 31P
$C_{124}.C_4^2$ 1A $1$ $1$ $C_{124}.C_4^2$ 1A 1A $1$
$C_{124}.C_4^2$ 2A $2$ $1$ $C_{124}.C_4^2$ 1A 2A $c$
$C_{124}.C_4^2$ 2B $2$ $1$ $C_{124}.C_4^2$ 1A 2B $b^{124}c$
$C_{124}.C_4^2$ 2C $2$ $1$ $C_{124}.C_4^2$ 1A 2C $b^{124}$
$C_{124}.C_4^2$ 2D $2$ $2$ $C_2^2\times C_{248}$ 1A 2D $a^{2}b^{186}$
$C_{124}.C_4^2$ 2E $2$ $2$ $C_2^2\times C_{248}$ 1A 2E $a^{2}b^{186}c$
$C_{124}.C_4^2$ 4A1 $4$ $1$ $C_{124}.C_4^2$ 2C 4A-1 $a^{2}b^{124}$
$C_{124}.C_4^2$ 4A-1 $4$ $1$ $C_{124}.C_4^2$ 2C 4A1 $a^{2}$
$C_{124}.C_4^2$ 4B1 $4$ $1$ $C_{124}.C_4^2$ 2C 4B-1 $a^{2}b^{124}c$
$C_{124}.C_4^2$ 4B-1 $4$ $1$ $C_{124}.C_4^2$ 2C 4B1 $a^{2}c$
$C_{124}.C_4^2$ 4C $4$ $2$ $C_2^2\times C_{248}$ 2C 4C $b^{62}c$
$C_{124}.C_4^2$ 4D $4$ $2$ $C_2^2\times C_{248}$ 2C 4D $b^{62}$
$C_{124}.C_4^2$ 8A1 $8$ $2$ $C_2^2\times C_{248}$ 4D 8A3 $b^{31}$
$C_{124}.C_4^2$ 8A-1 $8$ $2$ $C_2^2\times C_{248}$ 4D 8A-3 $b^{217}$
$C_{124}.C_4^2$ 8A3 $8$ $2$ $C_2^2\times C_{248}$ 4D 8A1 $b^{93}$
$C_{124}.C_4^2$ 8A-3 $8$ $2$ $C_2^2\times C_{248}$ 4D 8A-1 $b^{155}$
$C_{124}.C_4^2$ 8B1 $8$ $2$ $C_2^2\times C_{248}$ 4D 8B3 $a^{2}b^{155}$
$C_{124}.C_4^2$ 8B-1 $8$ $2$ $C_2^2\times C_{248}$ 4D 8B-3 $a^{2}b^{217}$
$C_{124}.C_4^2$ 8B3 $8$ $2$ $C_2^2\times C_{248}$ 4D 8B1 $a^{2}b^{93}$
$C_{124}.C_4^2$ 8B-3 $8$ $2$ $C_2^2\times C_{248}$ 4D 8B-1 $a^{2}b^{31}$
$C_{124}.C_4^2$ 8C1 $8$ $124$ $C_2\times C_8$ 4A1 8C-1 $a^{3}b^{226}$
$C_{124}.C_4^2$ 8C-1 $8$ $124$ $C_2\times C_8$ 4A-1 8C1 $ab^{102}$
$C_{124}.C_4^2$ 8D1 $8$ $124$ $C_2\times C_8$ 4B1 8D-1 $ab^{83}$
$C_{124}.C_4^2$ 8D-1 $8$ $124$ $C_2\times C_8$ 4B-1 8D1 $a^{3}b^{83}c$
$C_{124}.C_4^2$ 8E1 $8$ $124$ $C_2\times C_8$ 4B-1 8E-1 $a^{3}b^{97}c$
$C_{124}.C_4^2$ 8E-1 $8$ $124$ $C_2\times C_8$ 4B1 8E1 $ab^{97}$
$C_{124}.C_4^2$ 8F1 $8$ $124$ $C_2\times C_8$ 4A1 8F-1 $a^{3}b^{224}$
$C_{124}.C_4^2$ 8F-1 $8$ $124$ $C_2\times C_8$ 4A-1 8F1 $ab^{100}$
$C_{124}.C_4^2$ 31A1 $31$ $2$ $C_2^2\times C_{248}$ 31A2 31A3 $b^{16}$
$C_{124}.C_4^2$ 31A2 $31$ $2$ $C_2^2\times C_{248}$ 31A4 31A6 $b^{32}$
$C_{124}.C_4^2$ 31A3 $31$ $2$ $C_2^2\times C_{248}$ 31A6 31A9 $b^{48}$
$C_{124}.C_4^2$ 31A4 $31$ $2$ $C_2^2\times C_{248}$ 31A8 31A12 $b^{64}$
$C_{124}.C_4^2$ 31A5 $31$ $2$ $C_2^2\times C_{248}$ 31A10 31A15 $b^{80}$
$C_{124}.C_4^2$ 31A6 $31$ $2$ $C_2^2\times C_{248}$ 31A12 31A13 $b^{96}$
$C_{124}.C_4^2$ 31A7 $31$ $2$ $C_2^2\times C_{248}$ 31A14 31A10 $b^{112}$
$C_{124}.C_4^2$ 31A8 $31$ $2$ $C_2^2\times C_{248}$ 31A15 31A7 $b^{128}$
$C_{124}.C_4^2$ 31A9 $31$ $2$ $C_2^2\times C_{248}$ 31A13 31A4 $b^{144}$
$C_{124}.C_4^2$ 31A10 $31$ $2$ $C_2^2\times C_{248}$ 31A11 31A1 $b^{160}$
$C_{124}.C_4^2$ 31A11 $31$ $2$ $C_2^2\times C_{248}$ 31A9 31A2 $b^{176}$
$C_{124}.C_4^2$ 31A12 $31$ $2$ $C_2^2\times C_{248}$ 31A7 31A5 $b^{192}$
$C_{124}.C_4^2$ 31A13 $31$ $2$ $C_2^2\times C_{248}$ 31A5 31A8 $b^{208}$
$C_{124}.C_4^2$ 31A14 $31$ $2$ $C_2^2\times C_{248}$ 31A3 31A11 $b^{224}$
$C_{124}.C_4^2$ 31A15 $31$ $2$ $C_2^2\times C_{248}$ 31A1 31A14 $b^{240}$
$C_{124}.C_4^2$ 62A1 $62$ $2$ $C_2^2\times C_{248}$ 31A1 62A3 $b^{8}c$
$C_{124}.C_4^2$ 62A3 $62$ $2$ $C_2^2\times C_{248}$ 31A3 62A9 $b^{24}c$
$C_{124}.C_4^2$ 62A5 $62$ $2$ $C_2^2\times C_{248}$ 31A5 62A15 $b^{40}c$
$C_{124}.C_4^2$ 62A7 $62$ $2$ $C_2^2\times C_{248}$ 31A7 62A21 $b^{56}c$
$C_{124}.C_4^2$ 62A9 $62$ $2$ $C_2^2\times C_{248}$ 31A9 62A27 $b^{72}c$
$C_{124}.C_4^2$ 62A11 $62$ $2$ $C_2^2\times C_{248}$ 31A11 62A29 $b^{88}c$
$C_{124}.C_4^2$ 62A13 $62$ $2$ $C_2^2\times C_{248}$ 31A13 62A23 $b^{104}c$
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