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Elements of the group are displayed as words in the presentation $\langle a, b \mid a^{6}=b^{54}=1, b^{a}=b^{19} \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P
$C_{54}:C_6$ 1A $1$ $1$ $C_{54}:C_6$ 1A 1A $1$
$C_{54}:C_6$ 2A $2$ $1$ $C_{54}:C_6$ 1A 2A $a^{3}$
$C_{54}:C_6$ 2B $2$ $1$ $C_{54}:C_6$ 1A 2B $b^{27}$
$C_{54}:C_6$ 2C $2$ $1$ $C_{54}:C_6$ 1A 2C $a^{3}b^{27}$
$C_{54}:C_6$ 3A1 $3$ $1$ $C_{54}:C_6$ 3A-1 1A $b^{36}$
$C_{54}:C_6$ 3A-1 $3$ $1$ $C_{54}:C_6$ 3A1 1A $b^{18}$
$C_{54}:C_6$ 3B1 $3$ $3$ $C_6\times C_{18}$ 3B-1 1A $a^{2}$
$C_{54}:C_6$ 3B-1 $3$ $3$ $C_6\times C_{18}$ 3B1 1A $a^{4}$
$C_{54}:C_6$ 6A1 $6$ $1$ $C_{54}:C_6$ 3A1 2A $a^{3}b^{18}$
$C_{54}:C_6$ 6A-1 $6$ $1$ $C_{54}:C_6$ 3A-1 2A $a^{3}b^{36}$
$C_{54}:C_6$ 6B1 $6$ $1$ $C_{54}:C_6$ 3A-1 2B $b^{9}$
$C_{54}:C_6$ 6B-1 $6$ $1$ $C_{54}:C_6$ 3A1 2B $b^{45}$
$C_{54}:C_6$ 6C1 $6$ $1$ $C_{54}:C_6$ 3A-1 2C $a^{3}b^{9}$
$C_{54}:C_6$ 6C-1 $6$ $1$ $C_{54}:C_6$ 3A1 2C $a^{3}b^{45}$
$C_{54}:C_6$ 6D1 $6$ $3$ $C_6\times C_{18}$ 3B1 2A $a$
$C_{54}:C_6$ 6D-1 $6$ $3$ $C_6\times C_{18}$ 3B-1 2A $a^{5}$
$C_{54}:C_6$ 6E1 $6$ $3$ $C_6\times C_{18}$ 3B-1 2B $a^{2}b^{9}$
$C_{54}:C_6$ 6E-1 $6$ $3$ $C_6\times C_{18}$ 3B1 2B $a^{4}b^{9}$
$C_{54}:C_6$ 6F1 $6$ $3$ $C_6\times C_{18}$ 3B1 2C $ab^{9}$
$C_{54}:C_6$ 6F-1 $6$ $3$ $C_6\times C_{18}$ 3B-1 2C $a^{5}b^{9}$
$C_{54}:C_6$ 9A1 $9$ $1$ $C_{54}:C_6$ 9A2 3A-1 $b^{6}$
$C_{54}:C_6$ 9A-1 $9$ $1$ $C_{54}:C_6$ 9A-2 3A1 $b^{48}$
$C_{54}:C_6$ 9A2 $9$ $1$ $C_{54}:C_6$ 9A4 3A1 $b^{12}$
$C_{54}:C_6$ 9A-2 $9$ $1$ $C_{54}:C_6$ 9A-4 3A-1 $b^{42}$
$C_{54}:C_6$ 9A4 $9$ $1$ $C_{54}:C_6$ 9A-1 3A-1 $b^{24}$
$C_{54}:C_6$ 9A-4 $9$ $1$ $C_{54}:C_6$ 9A1 3A1 $b^{30}$
$C_{54}:C_6$ 9B1 $9$ $3$ $C_6\times C_{18}$ 9B-1 3A1 $a^{2}b^{12}$
$C_{54}:C_6$ 9B-1 $9$ $3$ $C_6\times C_{18}$ 9B1 3A-1 $a^{4}b^{42}$
$C_{54}:C_6$ 9C1 $9$ $3$ $C_6\times C_{18}$ 9C-1 3A-1 $a^{2}b^{24}$
$C_{54}:C_6$ 9C-1 $9$ $3$ $C_6\times C_{18}$ 9C1 3A1 $a^{4}b^{30}$
$C_{54}:C_6$ 18A1 $18$ $1$ $C_{54}:C_6$ 9A1 6B1 $b^{3}$
$C_{54}:C_6$ 18A-1 $18$ $1$ $C_{54}:C_6$ 9A-1 6B-1 $b^{51}$
$C_{54}:C_6$ 18A5 $18$ $1$ $C_{54}:C_6$ 9A-4 6B-1 $b^{15}$
$C_{54}:C_6$ 18A-5 $18$ $1$ $C_{54}:C_6$ 9A4 6B1 $b^{39}$
$C_{54}:C_6$ 18A7 $18$ $1$ $C_{54}:C_6$ 9A-2 6B1 $b^{21}$
$C_{54}:C_6$ 18A-7 $18$ $1$ $C_{54}:C_6$ 9A2 6B-1 $b^{33}$
$C_{54}:C_6$ 18B1 $18$ $1$ $C_{54}:C_6$ 9A2 6A1 $a^{3}b^{6}$
$C_{54}:C_6$ 18B-1 $18$ $1$ $C_{54}:C_6$ 9A-2 6A-1 $a^{3}b^{48}$
$C_{54}:C_6$ 18B5 $18$ $1$ $C_{54}:C_6$ 9A1 6A-1 $a^{3}b^{30}$
$C_{54}:C_6$ 18B-5 $18$ $1$ $C_{54}:C_6$ 9A-1 6A1 $a^{3}b^{24}$
$C_{54}:C_6$ 18B7 $18$ $1$ $C_{54}:C_6$ 9A-4 6A1 $a^{3}b^{42}$
$C_{54}:C_6$ 18B-7 $18$ $1$ $C_{54}:C_6$ 9A4 6A-1 $a^{3}b^{12}$
$C_{54}:C_6$ 18C1 $18$ $1$ $C_{54}:C_6$ 9A1 6C1 $a^{3}b^{3}$
$C_{54}:C_6$ 18C-1 $18$ $1$ $C_{54}:C_6$ 9A-1 6C-1 $a^{3}b^{51}$
$C_{54}:C_6$ 18C5 $18$ $1$ $C_{54}:C_6$ 9A-4 6C-1 $a^{3}b^{15}$
$C_{54}:C_6$ 18C-5 $18$ $1$ $C_{54}:C_6$ 9A4 6C1 $a^{3}b^{39}$
$C_{54}:C_6$ 18C7 $18$ $1$ $C_{54}:C_6$ 9A-2 6C1 $a^{3}b^{21}$
$C_{54}:C_6$ 18C-7 $18$ $1$ $C_{54}:C_6$ 9A2 6C-1 $a^{3}b^{33}$
$C_{54}:C_6$ 18D1 $18$ $3$ $C_6\times C_{18}$ 9B1 6A1 $ab^{6}$
$C_{54}:C_6$ 18D-1 $18$ $3$ $C_6\times C_{18}$ 9B-1 6A-1 $a^{5}b^{12}$
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