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Elements of the group are displayed as words in the presentation $\langle a, b, c \mid a^{27}=b^{2}=c^{2}=[b,c]=1, b^{a}=bc, c^{a}=b \rangle$ .

Group Label Order Size Centralizer Powers Representative
2P 3P
$C_2^2:C_{27}$ 1A $1$ $1$ $C_2^2:C_{27}$ 1A 1A $1$
$C_2^2:C_{27}$ 2A $2$ $3$ $C_2\times C_{18}$ 1A 2A $c$
$C_2^2:C_{27}$ 3A1 $3$ $1$ $C_2^2:C_{27}$ 3A-1 1A $a^{18}$
$C_2^2:C_{27}$ 3A-1 $3$ $1$ $C_2^2:C_{27}$ 3A1 1A $a^{9}$
$C_2^2:C_{27}$ 6A1 $6$ $3$ $C_2\times C_{18}$ 3A1 2A $a^{9}c$
$C_2^2:C_{27}$ 6A-1 $6$ $3$ $C_2\times C_{18}$ 3A-1 2A $a^{18}c$
$C_2^2:C_{27}$ 9A1 $9$ $1$ $C_2^2:C_{27}$ 9A2 3A1 $a^{6}$
$C_2^2:C_{27}$ 9A-1 $9$ $1$ $C_2^2:C_{27}$ 9A-2 3A-1 $a^{21}$
$C_2^2:C_{27}$ 9A2 $9$ $1$ $C_2^2:C_{27}$ 9A4 3A-1 $a^{12}$
$C_2^2:C_{27}$ 9A-2 $9$ $1$ $C_2^2:C_{27}$ 9A-4 3A1 $a^{15}$
$C_2^2:C_{27}$ 9A4 $9$ $1$ $C_2^2:C_{27}$ 9A-1 3A1 $a^{24}$
$C_2^2:C_{27}$ 9A-4 $9$ $1$ $C_2^2:C_{27}$ 9A1 3A-1 $a^{3}$
$C_2^2:C_{27}$ 18A1 $18$ $3$ $C_2\times C_{18}$ 9A1 6A1 $a^{3}c$
$C_2^2:C_{27}$ 18A-1 $18$ $3$ $C_2\times C_{18}$ 9A-1 6A-1 $a^{24}c$
$C_2^2:C_{27}$ 18A5 $18$ $3$ $C_2\times C_{18}$ 9A-4 6A-1 $a^{15}c$
$C_2^2:C_{27}$ 18A-5 $18$ $3$ $C_2\times C_{18}$ 9A4 6A1 $a^{12}c$
$C_2^2:C_{27}$ 18A7 $18$ $3$ $C_2\times C_{18}$ 9A-2 6A1 $a^{21}c$
$C_2^2:C_{27}$ 18A-7 $18$ $3$ $C_2\times C_{18}$ 9A2 6A-1 $a^{6}c$
$C_2^2:C_{27}$ 27A1 $27$ $4$ $C_{27}$ 27A2 9A-4 $a$
$C_2^2:C_{27}$ 27A-1 $27$ $4$ $C_{27}$ 27A-2 9A4 $a^{26}$
$C_2^2:C_{27}$ 27A2 $27$ $4$ $C_{27}$ 27A4 9A1 $a^{2}$
$C_2^2:C_{27}$ 27A-2 $27$ $4$ $C_{27}$ 27A-4 9A-1 $a^{25}$
$C_2^2:C_{27}$ 27A4 $27$ $4$ $C_{27}$ 27A8 9A2 $a^{4}$
$C_2^2:C_{27}$ 27A-4 $27$ $4$ $C_{27}$ 27A-8 9A-2 $a^{23}$
$C_2^2:C_{27}$ 27A5 $27$ $4$ $C_{27}$ 27A10 9A-2 $a^{5}$
$C_2^2:C_{27}$ 27A-5 $27$ $4$ $C_{27}$ 27A-10 9A2 $a^{22}$
$C_2^2:C_{27}$ 27A7 $27$ $4$ $C_{27}$ 27A-13 9A-1 $a^{7}$
$C_2^2:C_{27}$ 27A-7 $27$ $4$ $C_{27}$ 27A13 9A1 $a^{20}$
$C_2^2:C_{27}$ 27A8 $27$ $4$ $C_{27}$ 27A-11 9A4 $a^{8}$
$C_2^2:C_{27}$ 27A-8 $27$ $4$ $C_{27}$ 27A11 9A-4 $a^{19}$
$C_2^2:C_{27}$ 27A10 $27$ $4$ $C_{27}$ 27A-7 9A-4 $a^{10}$
$C_2^2:C_{27}$ 27A-10 $27$ $4$ $C_{27}$ 27A7 9A4 $a^{17}$
$C_2^2:C_{27}$ 27A11 $27$ $4$ $C_{27}$ 27A-5 9A1 $a^{11}$
$C_2^2:C_{27}$ 27A-11 $27$ $4$ $C_{27}$ 27A5 9A-1 $a^{16}$
$C_2^2:C_{27}$ 27A13 $27$ $4$ $C_{27}$ 27A-1 9A2 $a^{13}$
$C_2^2:C_{27}$ 27A-13 $27$ $4$ $C_{27}$ 27A1 9A-2 $a^{14}$
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