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Elements of the group are displayed as words in the presentation $\langle a, b, c, d, e, f \mid b^{6}=c^{18}=d^{18}=e^{9}=f^{9}=[c,e]=[e,f]=1, a^{2}=e^{8}f^{5}, b^{a}=b^{2}c^{9}d^{13}e^{3}f^{5}, c^{a}=b^{3}c^{16}d^{11}e^{8}f^{6}, d^{a}=c^{14}d^{9}e^{6}f, e^{a}=f^{4}, f^{a}=e^{7}, c^{b}=c^{5}d^{12}e^{6}f^{8}, d^{b}=d^{7}f^{6}, e^{b}=e^{4}, f^{b}=f^{2}, d^{c}=d^{17}e^{6}f^{4}, f^{c}=f^{8}, e^{d}=e^{8}, f^{d}=f^{8} \rangle$ .
| Group | Label | Order | Size | Centralizer | Powers | Representative | |
|---|---|---|---|---|---|---|---|
| 2P | 3P | ||||||
| $C_9^4.C_6.C_2^3$ | 1A | $1$ | $1$ | not computed | 1A | 1A | $1$ |
| $C_9^4.C_6.C_2^3$ | 2A | $2$ | $81$ | not computed | 1A | 2A | $d^{9}e^{6}f^{8}$ |
| $C_9^4.C_6.C_2^3$ | 2B | $2$ | $81$ | not computed | 1A | 2B | $b^{3}c^{9}d^{12}f$ |
| $C_9^4.C_6.C_2^3$ | 2C | $2$ | $162$ | not computed | 1A | 2C | $c^{9}d^{9}e^{6}f^{2}$ |
| $C_9^4.C_6.C_2^3$ | 2D | $2$ | $162$ | not computed | 1A | 2D | $b^{3}d^{9}e^{4}f^{3}$ |
| $C_9^4.C_6.C_2^3$ | 2E | $2$ | $486$ | not computed | 1A | 2E | $ac^{16}d^{13}e^{7}f^{3}$ |
| $C_9^4.C_6.C_2^3$ | 2F | $2$ | $486$ | not computed | 1A | 2F | $ab^{5}c^{5}d^{17}e^{4}f^{5}$ |
| $C_9^4.C_6.C_2^3$ | 2G | $2$ | $6561$ | not computed | 1A | 2G | $b^{3}c^{3}d^{11}e^{3}f^{4}$ |
| $C_9^4.C_6.C_2^3$ | 3A | $3$ | $4$ | not computed | 3A | 1A | $c^{6}$ |
| $C_9^4.C_6.C_2^3$ | 3B | $3$ | $4$ | not computed | 3B | 1A | $f^{3}$ |
| $C_9^4.C_6.C_2^3$ | 3C | $3$ | $4$ | not computed | 3C | 1A | $c^{6}d^{6}$ |
| $C_9^4.C_6.C_2^3$ | 3D | $3$ | $4$ | not computed | 3D | 1A | $e^{3}f^{3}$ |
| $C_9^4.C_6.C_2^3$ | 3E | $3$ | $8$ | not computed | 3E | 1A | $d^{6}f^{3}$ |
| $C_9^4.C_6.C_2^3$ | 3F | $3$ | $8$ | not computed | 3F | 1A | $c^{6}f^{3}$ |
| $C_9^4.C_6.C_2^3$ | 3G | $3$ | $8$ | not computed | 3G | 1A | $c^{6}d^{6}e^{3}f^{3}$ |
| $C_9^4.C_6.C_2^3$ | 3H | $3$ | $8$ | not computed | 3H | 1A | $c^{6}d^{6}e^{3}f^{6}$ |
| $C_9^4.C_6.C_2^3$ | 3I | $3$ | $16$ | not computed | 3I | 1A | $d^{6}e^{3}f^{3}$ |
| $C_9^4.C_6.C_2^3$ | 3J | $3$ | $16$ | not computed | 3J | 1A | $c^{6}d^{6}f^{3}$ |
| $C_9^4.C_6.C_2^3$ | 3K | $3$ | $162$ | not computed | 3K | 1A | $b^{4}d^{6}f^{3}$ |
| $C_9^4.C_6.C_2^3$ | 4A | $4$ | $39366$ | not computed | 2G | 4A | $ab^{5}c^{8}d^{14}e^{2}f^{5}$ |
| $C_9^4.C_6.C_2^3$ | 4B | $4$ | $39366$ | not computed | 2G | 4B | $ab^{2}cd^{12}e^{6}f^{4}$ |
| $C_9^4.C_6.C_2^3$ | 6A | $6$ | $324$ | not computed | 3B | 2C | $c^{9}d^{9}e^{6}f^{5}$ |
| $C_9^4.C_6.C_2^3$ | 6B | $6$ | $324$ | not computed | 3A | 2A | $d^{3}e^{6}f^{8}$ |
| $C_9^4.C_6.C_2^3$ | 6C | $6$ | $324$ | not computed | 3B | 2D | $c^{9}d^{4}e^{3}f^{8}$ |
| $C_9^4.C_6.C_2^3$ | 6D | $6$ | $324$ | not computed | 3B | 2B | $b^{3}c^{9}d^{12}e^{3}f$ |
| $C_9^4.C_6.C_2^3$ | 6E | $6$ | $324$ | not computed | 3A | 2C | $b^{3}c^{2}d^{12}e^{3}f^{7}$ |
| $C_9^4.C_6.C_2^3$ | 6F | $6$ | $324$ | not computed | 3A | 2D | $b^{3}c^{8}d^{15}e^{5}f^{3}$ |
| $C_9^4.C_6.C_2^3$ | 6G | $6$ | $324$ | not computed | 3D | 2B | $b^{3}c^{7}d^{14}e^{3}f^{7}$ |
| $C_9^4.C_6.C_2^3$ | 6H | $6$ | $324$ | not computed | 3C | 2A | $c^{6}d^{3}ef$ |
| $C_9^4.C_6.C_2^3$ | 6I | $6$ | $648$ | not computed | 3E | 2D | $b^{3}d^{15}e^{4}$ |
| $C_9^4.C_6.C_2^3$ | 6J | $6$ | $648$ | not computed | 3F | 2C | $c^{15}d^{17}e^{7}f^{5}$ |
| $C_9^4.C_6.C_2^3$ | 6K | $6$ | $972$ | not computed | 3C | 2E | $ac^{10}de^{7}f^{3}$ |
| $C_9^4.C_6.C_2^3$ | 6L | $6$ | $972$ | not computed | 3D | 2E | $ab^{5}c^{15}d^{16}e^{4}f$ |
| $C_9^4.C_6.C_2^3$ | 6M | $6$ | $972$ | not computed | 3D | 2F | $ab^{5}c^{5}d^{17}e^{5}f^{6}$ |
| $C_9^4.C_6.C_2^3$ | 6N | $6$ | $972$ | not computed | 3C | 2F | $abcd^{3}e^{4}f^{8}$ |
| $C_9^4.C_6.C_2^3$ | 6O | $6$ | $1458$ | not computed | 3K | 2A | $b^{4}d^{15}e^{7}f^{3}$ |
| $C_9^4.C_6.C_2^3$ | 6P | $6$ | $1458$ | not computed | 3K | 2B | $b^{5}c^{9}d^{16}e^{3}f^{4}$ |
| $C_9^4.C_6.C_2^3$ | 6Q1 | $6$ | $1458$ | not computed | 3K | 2C | $b^{2}c^{9}d^{9}e^{8}f^{2}$ |
| $C_9^4.C_6.C_2^3$ | 6Q-1 | $6$ | $1458$ | not computed | 3K | 2C | $b^{4}c^{9}d^{3}e^{5}f^{2}$ |
| $C_9^4.C_6.C_2^3$ | 6R1 | $6$ | $1458$ | not computed | 3K | 2D | $b^{5}c^{2}d^{3}e^{4}f^{6}$ |
| $C_9^4.C_6.C_2^3$ | 6R-1 | $6$ | $1458$ | not computed | 3K | 2D | $bc^{8}d^{15}e^{7}$ |
| $C_9^4.C_6.C_2^3$ | 6S | $6$ | $1944$ | not computed | 3H | 2F | $ab^{5}c^{11}d^{5}e^{7}f^{2}$ |
| $C_9^4.C_6.C_2^3$ | 6T | $6$ | $1944$ | not computed | 3G | 2E | $ab^{5}c^{15}d^{4}e^{7}f^{7}$ |
| $C_9^4.C_6.C_2^3$ | 6U | $6$ | $13122$ | not computed | 3K | 2G | $b^{5}c^{3}d^{17}e^{3}f^{7}$ |
| $C_9^4.C_6.C_2^3$ | 9A | $9$ | $12$ | not computed | 9A | 3B | $f^{2}$ |
| $C_9^4.C_6.C_2^3$ | 9B | $9$ | $12$ | not computed | 9B | 3A | $d^{4}$ |
| $C_9^4.C_6.C_2^3$ | 9C1 | $9$ | $12$ | not computed | 9C2 | 3D | $e^{7}f^{4}$ |
| $C_9^4.C_6.C_2^3$ | 9C2 | $9$ | $12$ | not computed | 9C4 | 3D | $e^{5}f^{8}$ |
| $C_9^4.C_6.C_2^3$ | 9C4 | $9$ | $12$ | not computed | 9C1 | 3D | $ef^{7}$ |
| $C_9^4.C_6.C_2^3$ | 9D1 | $9$ | $12$ | not computed | 9D2 | 3C | $c^{2}d^{2}e^{3}$ |