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Elements of the group are displayed as words in the presentation $\langle a, b \mid a^{22}=b^{69}=1, b^{a}=b^{43} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P 11P 23P
$C_3\times F_{23}$ 1A $1$ $1$ $C_3\times F_{23}$ 1A 1A 1A 1A $1$
$C_3\times F_{23}$ 2A $2$ $23$ $C_{66}$ 1A 2A 2A 2A $a^{11}b^{12}$
$C_3\times F_{23}$ 3A1 $3$ $1$ $C_3\times F_{23}$ 3A-1 1A 3A-1 3A-1 $b^{46}$
$C_3\times F_{23}$ 3A-1 $3$ $1$ $C_3\times F_{23}$ 3A1 1A 3A1 3A1 $b^{23}$
$C_3\times F_{23}$ 6A1 $6$ $23$ $C_{66}$ 3A1 2A 6A-1 6A-1 $a^{11}b^{35}$
$C_3\times F_{23}$ 6A-1 $6$ $23$ $C_{66}$ 3A-1 2A 6A1 6A1 $a^{11}b^{58}$
$C_3\times F_{23}$ 11A1 $11$ $23$ $C_{66}$ 11A2 11A3 11A5 1A $a^{6}b^{48}$
$C_3\times F_{23}$ 11A-1 $11$ $23$ $C_{66}$ 11A-2 11A-3 11A-5 1A $a^{16}b^{66}$
$C_3\times F_{23}$ 11A2 $11$ $23$ $C_{66}$ 11A4 11A-5 11A-1 1A $a^{12}b^{57}$
$C_3\times F_{23}$ 11A-2 $11$ $23$ $C_{66}$ 11A-4 11A5 11A1 1A $a^{10}b^{27}$
$C_3\times F_{23}$ 11A3 $11$ $23$ $C_{66}$ 11A-5 11A-2 11A4 1A $a^{18}b^{63}$
$C_3\times F_{23}$ 11A-3 $11$ $23$ $C_{66}$ 11A5 11A2 11A-4 1A $a^{4}b^{3}$
$C_3\times F_{23}$ 11A4 $11$ $23$ $C_{66}$ 11A-3 11A1 11A-2 1A $a^{2}b^{21}$
$C_3\times F_{23}$ 11A-4 $11$ $23$ $C_{66}$ 11A3 11A-1 11A2 1A $a^{20}b^{36}$
$C_3\times F_{23}$ 11A5 $11$ $23$ $C_{66}$ 11A-1 11A4 11A3 1A $a^{8}b^{39}$
$C_3\times F_{23}$ 11A-5 $11$ $23$ $C_{66}$ 11A1 11A-4 11A-3 1A $a^{14}b^{51}$
$C_3\times F_{23}$ 22A1 $22$ $23$ $C_{66}$ 11A1 22A3 22A5 2A $a^{3}b^{30}$
$C_3\times F_{23}$ 22A-1 $22$ $23$ $C_{66}$ 11A-1 22A-3 22A-5 2A $a^{19}b^{42}$
$C_3\times F_{23}$ 22A3 $22$ $23$ $C_{66}$ 11A3 22A9 22A-7 2A $a^{9}b^{45}$
$C_3\times F_{23}$ 22A-3 $22$ $23$ $C_{66}$ 11A-3 22A-9 22A7 2A $a^{13}b^{60}$
$C_3\times F_{23}$ 22A5 $22$ $23$ $C_{66}$ 11A5 22A-7 22A3 2A $a^{15}b^{9}$
$C_3\times F_{23}$ 22A-5 $22$ $23$ $C_{66}$ 11A-5 22A7 22A-3 2A $a^{7}b^{18}$
$C_3\times F_{23}$ 22A7 $22$ $23$ $C_{66}$ 11A-4 22A-1 22A-9 2A $a^{21}b^{54}$
$C_3\times F_{23}$ 22A-7 $22$ $23$ $C_{66}$ 11A4 22A1 22A9 2A $ab^{24}$
$C_3\times F_{23}$ 22A9 $22$ $23$ $C_{66}$ 11A-2 22A5 22A1 2A $a^{5}b^{15}$
$C_3\times F_{23}$ 22A-9 $22$ $23$ $C_{66}$ 11A2 22A-5 22A-1 2A $a^{17}b^{33}$
$C_3\times F_{23}$ 23A $23$ $22$ $C_{69}$ 23A 23A 23A 23A $b^{3}$
$C_3\times F_{23}$ 33A1 $33$ $23$ $C_{66}$ 33A2 11A1 33A5 3A1 $a^{2}b^{44}$
$C_3\times F_{23}$ 33A-1 $33$ $23$ $C_{66}$ 33A-2 11A-1 33A-5 3A-1 $a^{20}b^{13}$
$C_3\times F_{23}$ 33A2 $33$ $23$ $C_{66}$ 33A4 11A2 33A10 3A-1 $a^{4}b^{49}$
$C_3\times F_{23}$ 33A-2 $33$ $23$ $C_{66}$ 33A-4 11A-2 33A-10 3A1 $a^{18}b^{17}$
$C_3\times F_{23}$ 33A4 $33$ $23$ $C_{66}$ 33A8 11A4 33A-13 3A1 $a^{8}b^{62}$
$C_3\times F_{23}$ 33A-4 $33$ $23$ $C_{66}$ 33A-8 11A-4 33A13 3A-1 $a^{14}b^{28}$
$C_3\times F_{23}$ 33A5 $33$ $23$ $C_{66}$ 33A10 11A5 33A-8 3A-1 $a^{10}b^{4}$
$C_3\times F_{23}$ 33A-5 $33$ $23$ $C_{66}$ 33A-10 11A-5 33A8 3A1 $a^{12}b^{11}$
$C_3\times F_{23}$ 33A7 $33$ $23$ $C_{66}$ 33A14 11A-4 33A2 3A1 $a^{14}b^{5}$
$C_3\times F_{23}$ 33A-7 $33$ $23$ $C_{66}$ 33A-14 11A4 33A-2 3A-1 $a^{8}b^{16}$
$C_3\times F_{23}$ 33A8 $33$ $23$ $C_{66}$ 33A16 11A-3 33A7 3A-1 $a^{16}b^{43}$
$C_3\times F_{23}$ 33A-8 $33$ $23$ $C_{66}$ 33A-16 11A3 33A-7 3A1 $a^{6}b^{2}$
$C_3\times F_{23}$ 33A10 $33$ $23$ $C_{66}$ 33A-13 11A-1 33A-16 3A1 $a^{20}b^{59}$
$C_3\times F_{23}$ 33A-10 $33$ $23$ $C_{66}$ 33A13 11A1 33A16 3A-1 $a^{2}b^{67}$
$C_3\times F_{23}$ 33A13 $33$ $23$ $C_{66}$ 33A-7 11A2 33A-1 3A1 $a^{4}b^{26}$
$C_3\times F_{23}$ 33A-13 $33$ $23$ $C_{66}$ 33A7 11A-2 33A1 3A-1 $a^{18}b^{40}$
$C_3\times F_{23}$ 33A14 $33$ $23$ $C_{66}$ 33A-5 11A3 33A4 3A-1 $a^{6}b^{25}$
$C_3\times F_{23}$ 33A-14 $33$ $23$ $C_{66}$ 33A5 11A-3 33A-4 3A1 $a^{16}b^{20}$
$C_3\times F_{23}$ 33A16 $33$ $23$ $C_{66}$ 33A-1 11A5 33A14 3A1 $a^{10}b^{50}$
$C_3\times F_{23}$ 33A-16 $33$ $23$ $C_{66}$ 33A1 11A-5 33A-14 3A-1 $a^{12}b^{34}$
$C_3\times F_{23}$ 66A1 $66$ $23$ $C_{66}$ 33A1 22A1 66A5 6A1 $ab$
$C_3\times F_{23}$ 66A-1 $66$ $23$ $C_{66}$ 33A-1 22A-1 66A-5 6A-1 $a^{21}b^{2}$
$C_3\times F_{23}$ 66A5 $66$ $23$ $C_{66}$ 33A5 22A5 66A25 6A-1 $a^{5}b^{2}$
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