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Elements of the group are displayed as words in the presentation $\langle a, b \mid a^{4}=b^{33}=1, b^{a}=b^{23} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P 11P
$C_3:C_{44}$ 1A $1$ $1$ $C_3:C_{44}$ 1A 1A 1A $1$
$C_3:C_{44}$ 2A $2$ $1$ $C_3:C_{44}$ 1A 2A 2A $a^{2}$
$C_3:C_{44}$ 3A $3$ $2$ $C_{66}$ 3A 1A 3A $b^{22}$
$C_3:C_{44}$ 4A1 $4$ $3$ $C_{44}$ 2A 4A-1 4A1 $a^{3}$
$C_3:C_{44}$ 4A-1 $4$ $3$ $C_{44}$ 2A 4A1 4A-1 $a$
$C_3:C_{44}$ 6A $6$ $2$ $C_{66}$ 3A 2A 6A $a^{2}b^{11}$
$C_3:C_{44}$ 11A1 $11$ $1$ $C_3:C_{44}$ 11A2 11A3 11A5 $b^{12}$
$C_3:C_{44}$ 11A-1 $11$ $1$ $C_3:C_{44}$ 11A-2 11A-3 11A-5 $b^{21}$
$C_3:C_{44}$ 11A2 $11$ $1$ $C_3:C_{44}$ 11A4 11A-5 11A-1 $b^{24}$
$C_3:C_{44}$ 11A-2 $11$ $1$ $C_3:C_{44}$ 11A-4 11A5 11A1 $b^{9}$
$C_3:C_{44}$ 11A3 $11$ $1$ $C_3:C_{44}$ 11A-5 11A-2 11A4 $b^{3}$
$C_3:C_{44}$ 11A-3 $11$ $1$ $C_3:C_{44}$ 11A5 11A2 11A-4 $b^{30}$
$C_3:C_{44}$ 11A4 $11$ $1$ $C_3:C_{44}$ 11A-3 11A1 11A-2 $b^{15}$
$C_3:C_{44}$ 11A-4 $11$ $1$ $C_3:C_{44}$ 11A3 11A-1 11A2 $b^{18}$
$C_3:C_{44}$ 11A5 $11$ $1$ $C_3:C_{44}$ 11A-1 11A4 11A3 $b^{27}$
$C_3:C_{44}$ 11A-5 $11$ $1$ $C_3:C_{44}$ 11A1 11A-4 11A-3 $b^{6}$
$C_3:C_{44}$ 22A1 $22$ $1$ $C_3:C_{44}$ 11A1 22A3 22A5 $a^{2}b^{6}$
$C_3:C_{44}$ 22A-1 $22$ $1$ $C_3:C_{44}$ 11A-1 22A-3 22A-5 $a^{2}b^{27}$
$C_3:C_{44}$ 22A3 $22$ $1$ $C_3:C_{44}$ 11A3 22A9 22A-7 $a^{2}b^{18}$
$C_3:C_{44}$ 22A-3 $22$ $1$ $C_3:C_{44}$ 11A-3 22A-9 22A7 $a^{2}b^{15}$
$C_3:C_{44}$ 22A5 $22$ $1$ $C_3:C_{44}$ 11A5 22A-7 22A3 $a^{2}b^{30}$
$C_3:C_{44}$ 22A-5 $22$ $1$ $C_3:C_{44}$ 11A-5 22A7 22A-3 $a^{2}b^{3}$
$C_3:C_{44}$ 22A7 $22$ $1$ $C_3:C_{44}$ 11A-4 22A-1 22A-9 $a^{2}b^{9}$
$C_3:C_{44}$ 22A-7 $22$ $1$ $C_3:C_{44}$ 11A4 22A1 22A9 $a^{2}b^{24}$
$C_3:C_{44}$ 22A9 $22$ $1$ $C_3:C_{44}$ 11A-2 22A5 22A1 $a^{2}b^{21}$
$C_3:C_{44}$ 22A-9 $22$ $1$ $C_3:C_{44}$ 11A2 22A-5 22A-1 $a^{2}b^{12}$
$C_3:C_{44}$ 33A1 $33$ $2$ $C_{66}$ 33A2 11A-5 33A5 $b^{2}$
$C_3:C_{44}$ 33A-1 $33$ $2$ $C_{66}$ 33A-2 11A5 33A-5 $b^{31}$
$C_3:C_{44}$ 33A2 $33$ $2$ $C_{66}$ 33A4 11A1 33A-1 $b^{4}$
$C_3:C_{44}$ 33A-2 $33$ $2$ $C_{66}$ 33A-4 11A-1 33A1 $b^{29}$
$C_3:C_{44}$ 33A4 $33$ $2$ $C_{66}$ 33A8 11A2 33A-2 $b^{8}$
$C_3:C_{44}$ 33A-4 $33$ $2$ $C_{66}$ 33A-8 11A-2 33A2 $b^{25}$
$C_3:C_{44}$ 33A5 $33$ $2$ $C_{66}$ 33A-1 11A-3 33A-8 $b^{10}$
$C_3:C_{44}$ 33A-5 $33$ $2$ $C_{66}$ 33A1 11A3 33A8 $b^{23}$
$C_3:C_{44}$ 33A8 $33$ $2$ $C_{66}$ 33A5 11A4 33A-4 $b^{16}$
$C_3:C_{44}$ 33A-8 $33$ $2$ $C_{66}$ 33A-5 11A-4 33A4 $b^{17}$
$C_3:C_{44}$ 44A1 $44$ $3$ $C_{44}$ 22A1 44A3 44A5 $ab^{3}$
$C_3:C_{44}$ 44A-1 $44$ $3$ $C_{44}$ 22A-1 44A-3 44A-5 $a^{3}b^{30}$
$C_3:C_{44}$ 44A3 $44$ $3$ $C_{44}$ 22A3 44A9 44A15 $a^{3}b^{9}$
$C_3:C_{44}$ 44A-3 $44$ $3$ $C_{44}$ 22A-3 44A-9 44A-15 $ab^{2}$
$C_3:C_{44}$ 44A5 $44$ $3$ $C_{44}$ 22A5 44A15 44A-19 $ab^{15}$
$C_3:C_{44}$ 44A-5 $44$ $3$ $C_{44}$ 22A-5 44A-15 44A19 $a^{3}b^{18}$
$C_3:C_{44}$ 44A7 $44$ $3$ $C_{44}$ 22A7 44A21 44A-9 $a^{3}b^{21}$
$C_3:C_{44}$ 44A-7 $44$ $3$ $C_{44}$ 22A-7 44A-21 44A9 $ab$
$C_3:C_{44}$ 44A9 $44$ $3$ $C_{44}$ 22A9 44A-17 44A1 $ab^{27}$
$C_3:C_{44}$ 44A-9 $44$ $3$ $C_{44}$ 22A-9 44A17 44A-1 $a^{3}b^{6}$
$C_3:C_{44}$ 44A13 $44$ $3$ $C_{44}$ 22A-9 44A-5 44A21 $ab^{6}$
$C_3:C_{44}$ 44A-13 $44$ $3$ $C_{44}$ 22A9 44A5 44A-21 $a^{3}b^{27}$
$C_3:C_{44}$ 44A15 $44$ $3$ $C_{44}$ 22A-7 44A1 44A-13 $a^{3}b$
$C_3:C_{44}$ 44A-15 $44$ $3$ $C_{44}$ 22A7 44A-1 44A13 $ab^{21}$
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