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Elements of the group are displayed as words in the presentation $\langle a, b, c, d, e, f \mid b^{2}=c^{18}=d^{9}=e^{9}=f^{9}=[d,e]=[d,f]=[e,f]=1, a^{12}=c^{6}d^{3}e^{3}f^{3}, b^{a}=bc^{4}d^{3}e^{8}, c^{a}=bc^{7}d^{8}e^{2}f, d^{a}=c^{8}ef^{4}, e^{a}=c^{2}e^{8}f^{5}, f^{a}=c^{4}d^{5}e^{6}f, c^{b}=c^{17}d^{2}e^{7}, d^{b}=d^{8}, e^{b}=e^{8}, f^{b}=f^{8}, d^{c}=d^{8}, e^{c}=e^{8}, f^{c}=e^{5}f \rangle$ .
| Group | Label | Order | Size | Centralizer | Powers | Representative | |
|---|---|---|---|---|---|---|---|
| 2P | 3P | ||||||
| $C_9:D_9^3.C_6$ | 1A | $1$ | $1$ | not computed | 1A | 1A | $1$ |
| $C_9:D_9^3.C_6$ | 2A | $2$ | $81$ | not computed | 1A | 2A | $a^{6}c^{6}d^{3}e^{6}f^{3}$ |
| $C_9:D_9^3.C_6$ | 2B | $2$ | $81$ | not computed | 1A | 2B | $a^{6}bc^{6}d^{4}e^{2}$ |
| $C_9:D_9^3.C_6$ | 2C | $2$ | $162$ | not computed | 1A | 2C | $c^{9}d$ |
| $C_9:D_9^3.C_6$ | 2D | $2$ | $162$ | not computed | 1A | 2D | $a^{6}bc^{17}d^{4}e^{6}f$ |
| $C_9:D_9^3.C_6$ | 2E | $2$ | $6561$ | not computed | 1A | 2E | $bc^{4}d^{7}e^{2}f^{8}$ |
| $C_9:D_9^3.C_6$ | 3A | $3$ | $4$ | not computed | 3A | 1A | $e^{3}$ |
| $C_9:D_9^3.C_6$ | 3B | $3$ | $4$ | not computed | 3B | 1A | $d^{3}$ |
| $C_9:D_9^3.C_6$ | 3C | $3$ | $4$ | not computed | 3C | 1A | $c^{6}f^{6}$ |
| $C_9:D_9^3.C_6$ | 3D | $3$ | $4$ | not computed | 3D | 1A | $c^{6}d^{3}e^{3}f^{3}$ |
| $C_9:D_9^3.C_6$ | 3E | $3$ | $8$ | not computed | 3E | 1A | $e^{3}f^{3}$ |
| $C_9:D_9^3.C_6$ | 3F | $3$ | $8$ | not computed | 3F | 1A | $c^{6}f^{3}$ |
| $C_9:D_9^3.C_6$ | 3G | $3$ | $16$ | not computed | 3G | 1A | $f^{3}$ |
| $C_9:D_9^3.C_6$ | 3H | $3$ | $16$ | not computed | 3H | 1A | $d^{3}f^{3}$ |
| $C_9:D_9^3.C_6$ | 3I | $3$ | $16$ | not computed | 3I | 1A | $c^{6}d^{3}$ |
| $C_9:D_9^3.C_6$ | 3J1 | $3$ | $81$ | not computed | 3J-1 | 1A | $a^{4}c^{16}d^{2}e^{2}f^{5}$ |
| $C_9:D_9^3.C_6$ | 3J-1 | $3$ | $81$ | not computed | 3J1 | 1A | $a^{8}c^{2}d^{7}e^{7}f^{4}$ |
| $C_9:D_9^3.C_6$ | 4A1 | $4$ | $1458$ | not computed | 2A | 4A-1 | $a^{9}c^{8}d^{5}e^{4}f^{7}$ |
| $C_9:D_9^3.C_6$ | 4A-1 | $4$ | $1458$ | not computed | 2A | 4A1 | $a^{3}c^{14}d^{2}ef^{7}$ |
| $C_9:D_9^3.C_6$ | 4B1 | $4$ | $1458$ | not computed | 2B | 4B-1 | $a^{3}c^{5}d^{5}e^{6}f^{3}$ |
| $C_9:D_9^3.C_6$ | 4B-1 | $4$ | $1458$ | not computed | 2B | 4B1 | $a^{9}bc^{7}d^{5}e^{8}f$ |
| $C_9:D_9^3.C_6$ | 6A | $6$ | $324$ | not computed | 3B | 2C | $c^{15}de^{3}f^{3}$ |
| $C_9:D_9^3.C_6$ | 6B | $6$ | $324$ | not computed | 3D | 2A | $a^{6}c^{12}d^{6}f^{6}$ |
| $C_9:D_9^3.C_6$ | 6C | $6$ | $324$ | not computed | 3A | 2B | $a^{6}bc^{12}d^{4}e^{8}f^{6}$ |
| $C_9:D_9^3.C_6$ | 6D | $6$ | $324$ | not computed | 3A | 2D | $a^{6}bc^{11}d^{4}f^{4}$ |
| $C_9:D_9^3.C_6$ | 6E | $6$ | $324$ | not computed | 3B | 2A | $a^{6}d^{3}f^{3}$ |
| $C_9:D_9^3.C_6$ | 6F | $6$ | $324$ | not computed | 3C | 2B | $a^{6}bc^{4}d^{8}e^{4}f^{5}$ |
| $C_9:D_9^3.C_6$ | 6G | $6$ | $324$ | not computed | 3B | 2D | $a^{6}bc^{13}de^{4}f^{8}$ |
| $C_9:D_9^3.C_6$ | 6H | $6$ | $324$ | not computed | 3A | 2C | $c^{15}d^{2}ef^{6}$ |
| $C_9:D_9^3.C_6$ | 6I | $6$ | $648$ | not computed | 3F | 2D | $a^{6}bc^{13}de^{7}f^{5}$ |
| $C_9:D_9^3.C_6$ | 6J | $6$ | $648$ | not computed | 3E | 2C | $c^{9}df^{3}$ |
| $C_9:D_9^3.C_6$ | 6K1 | $6$ | $729$ | not computed | 3J1 | 2A | $a^{2}c^{4}de^{8}$ |
| $C_9:D_9^3.C_6$ | 6K-1 | $6$ | $729$ | not computed | 3J-1 | 2A | $a^{10}d^{2}e^{3}f^{4}$ |
| $C_9:D_9^3.C_6$ | 6L1 | $6$ | $729$ | not computed | 3J1 | 2B | $a^{2}bc^{16}d^{4}e^{7}f^{5}$ |
| $C_9:D_9^3.C_6$ | 6L-1 | $6$ | $729$ | not computed | 3J-1 | 2B | $a^{10}bc^{14}d^{6}e^{6}f^{4}$ |
| $C_9:D_9^3.C_6$ | 6M1 | $6$ | $1458$ | not computed | 3J1 | 2D | $a^{2}bc^{3}d^{5}e^{8}f^{6}$ |
| $C_9:D_9^3.C_6$ | 6M-1 | $6$ | $1458$ | not computed | 3J-1 | 2D | $a^{10}bcd^{6}e^{4}f^{8}$ |
| $C_9:D_9^3.C_6$ | 6N1 | $6$ | $1458$ | not computed | 3J1 | 2C | $a^{8}c^{5}d^{4}e^{6}f$ |
| $C_9:D_9^3.C_6$ | 6N-1 | $6$ | $1458$ | not computed | 3J-1 | 2C | $a^{4}c^{13}d^{6}e^{7}f^{8}$ |
| $C_9:D_9^3.C_6$ | 6O1 | $6$ | $6561$ | not computed | 3J-1 | 2E | $a^{4}bc^{12}d^{8}e^{6}f^{3}$ |
| $C_9:D_9^3.C_6$ | 6O-1 | $6$ | $6561$ | not computed | 3J1 | 2E | $a^{8}bc^{14}d^{6}e^{7}f^{4}$ |
| $C_9:D_9^3.C_6$ | 9A | $9$ | $12$ | not computed | 9A | 3A | $c^{8}e^{2}f^{8}$ |
| $C_9:D_9^3.C_6$ | 9B | $9$ | $12$ | not computed | 9B | 3D | $c^{4}de^{5}f^{2}$ |
| $C_9:D_9^3.C_6$ | 9C | $9$ | $12$ | not computed | 9C | 3C | $c^{14}d^{6}e^{7}f^{5}$ |
| $C_9:D_9^3.C_6$ | 9D | $9$ | $12$ | not computed | 9D | 3B | $c^{14}e^{4}f^{7}$ |
| $C_9:D_9^3.C_6$ | 9E | $9$ | $24$ | not computed | 9E | 3D | $c^{10}d^{5}e^{8}f^{5}$ |
| $C_9:D_9^3.C_6$ | 9F | $9$ | $24$ | not computed | 9F | 3B | $c^{2}ef^{4}$ |
| $C_9:D_9^3.C_6$ | 9G | $9$ | $24$ | not computed | 9G | 3E | $c^{8}$ |
| $C_9:D_9^3.C_6$ | 9H | $9$ | $24$ | not computed | 9H | 3E | $c^{6}e^{2}f^{8}$ |
| $C_9:D_9^3.C_6$ | 9I | $9$ | $24$ | not computed | 9I | 3B | $c^{10}d^{6}e^{8}f^{5}$ |