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Elements of the group are displayed as words in the presentation $\langle a, b, c, d, e, f, g, h \mid b^{2}=c^{2}=d^{4}=e^{9}=f^{18}=g^{9}=h^{9}=[b,c]=[b,e]=[c,e]=[e,g]=[e,h]=[g,h]=1, a^{6}=e^{4}f^{4}g^{7}h^{7}, b^{a}=cd^{2}e^{6}f^{10}gh^{8}, c^{a}=bcf^{8}g^{5}h^{7}, d^{a}=cd^{2}e^{2}f^{13}gh^{2}, e^{a}=e^{7}h^{7}, f^{a}=cd^{3}e^{4}fg^{3}h^{4}, g^{a}=f^{4}h^{8}, h^{a}=e^{2}f^{14}g^{2}h^{3}, d^{b}=d^{3}e^{5}f^{2}g^{5}h, f^{b}=d^{2}e^{4}f^{5}g^{3}h^{3}, g^{b}=ef^{2}h^{5}, h^{b}=e^{7}h^{8}, d^{c}=de^{6}h^{6}, f^{c}=d^{2}e^{4}f^{17}g^{2}h^{7}, g^{c}=ef^{16}h^{5}, h^{c}=e^{7}h^{8}, e^{d}=f^{2}g^{8}h^{4}, f^{d}=e^{4}f^{9}h^{5}, g^{d}=g^{8}h^{8}, h^{d}=g^{2}h, f^{e}=fh^{8}, g^{f}=g^{8}, h^{f}=h^{8} \rangle$ .
| Group | Label | Order | Size | Centralizer | Powers | Representative | |
|---|---|---|---|---|---|---|---|
| 2P | 3P | ||||||
| $D_9\wr C_2^2.C_3$ | 1A | $1$ | $1$ | not computed | 1A | 1A | $1$ |
| $D_9\wr C_2^2.C_3$ | 2A | $2$ | $36$ | not computed | 1A | 2A | $a^{3}ce^{2}f^{16}gh^{5}$ |
| $D_9\wr C_2^2.C_3$ | 2B | $2$ | $486$ | not computed | 1A | 2B | $bd^{2}ef^{2}g^{5}h$ |
| $D_9\wr C_2^2.C_3$ | 2C | $2$ | $972$ | not computed | 1A | 2C | $cd^{3}e^{4}f^{5}g^{2}h^{2}$ |
| $D_9\wr C_2^2.C_3$ | 2D | $2$ | $2916$ | not computed | 1A | 2D | $a^{3}bcd^{2}e^{2}f^{12}g^{3}h^{7}$ |
| $D_9\wr C_2^2.C_3$ | 2E | $2$ | $6561$ | not computed | 1A | 2E | $d^{2}e^{2}f^{2}g^{5}h^{2}$ |
| $D_9\wr C_2^2.C_3$ | 3A | $3$ | $8$ | not computed | 3A | 1A | $e^{3}$ |
| $D_9\wr C_2^2.C_3$ | 3B | $3$ | $16$ | not computed | 3B | 1A | $g^{3}$ |
| $D_9\wr C_2^2.C_3$ | 3C | $3$ | $24$ | not computed | 3C | 1A | $f^{6}$ |
| $D_9\wr C_2^2.C_3$ | 3D | $3$ | $32$ | not computed | 3D | 1A | $e^{3}g^{3}$ |
| $D_9\wr C_2^2.C_3$ | 3E1 | $3$ | $3888$ | not computed | 3E-1 | 1A | $a^{4}bcde^{4}f^{7}g^{3}$ |
| $D_9\wr C_2^2.C_3$ | 3E-1 | $3$ | $3888$ | not computed | 3E1 | 1A | $a^{2}e^{4}f^{9}g^{3}h^{8}$ |
| $D_9\wr C_2^2.C_3$ | 4A | $4$ | $17496$ | not computed | 2B | 4A | $a^{3}bcd^{3}e^{5}f^{5}g^{6}$ |
| $D_9\wr C_2^2.C_3$ | 4B | $4$ | $78732$ | not computed | 2E | 4B | $be^{6}f^{13}g^{7}h^{5}$ |
| $D_9\wr C_2^2.C_3$ | 6A | $6$ | $216$ | not computed | 3A | 2A | $a^{3}ce^{2}f^{4}g^{7}h^{8}$ |
| $D_9\wr C_2^2.C_3$ | 6B | $6$ | $288$ | not computed | 3D | 2A | $a^{3}e^{4}f^{4}g^{7}h^{7}$ |
| $D_9\wr C_2^2.C_3$ | 6C | $6$ | $432$ | not computed | 3C | 2A | $a^{3}ce^{5}f^{4}g^{7}h^{8}$ |
| $D_9\wr C_2^2.C_3$ | 6D | $6$ | $1944$ | not computed | 3C | 2B | $bd^{2}e^{4}f^{14}g^{8}h$ |
| $D_9\wr C_2^2.C_3$ | 6E | $6$ | $1944$ | not computed | 3A | 2B | $bce^{3}f^{6}g^{3}h^{6}$ |
| $D_9\wr C_2^2.C_3$ | 6F | $6$ | $3888$ | not computed | 3C | 2C | $cd^{3}e^{7}f^{11}g^{8}h^{5}$ |
| $D_9\wr C_2^2.C_3$ | 6G | $6$ | $3888$ | not computed | 3B | 2C | $bcd^{3}ef^{4}h^{4}$ |
| $D_9\wr C_2^2.C_3$ | 6H | $6$ | $5832$ | not computed | 3A | 2D | $a^{3}bcd^{2}e^{2}f^{6}g^{6}h^{4}$ |
| $D_9\wr C_2^2.C_3$ | 6I1 | $6$ | $11664$ | not computed | 3E-1 | 2A | $abcd^{2}e^{8}f^{9}g^{5}h^{7}$ |
| $D_9\wr C_2^2.C_3$ | 6I-1 | $6$ | $11664$ | not computed | 3E1 | 2A | $a^{5}bcd^{3}e^{7}f^{2}h^{7}$ |
| $D_9\wr C_2^2.C_3$ | 6J1 | $6$ | $34992$ | not computed | 3E1 | 2D | $a^{5}bcf^{13}g^{5}h^{3}$ |
| $D_9\wr C_2^2.C_3$ | 6J-1 | $6$ | $34992$ | not computed | 3E-1 | 2D | $abcde^{6}f^{15}g^{4}h^{7}$ |
| $D_9\wr C_2^2.C_3$ | 6K1 | $6$ | $104976$ | not computed | 3E1 | 2E | $a^{2}d^{2}e^{7}f^{11}g^{7}h^{6}$ |
| $D_9\wr C_2^2.C_3$ | 6K-1 | $6$ | $104976$ | not computed | 3E-1 | 2E | $a^{4}bcd^{3}e^{7}f^{13}g^{7}h^{5}$ |
| $D_9\wr C_2^2.C_3$ | 9A | $9$ | $24$ | not computed | 9A | 3A | $f^{4}g^{7}h^{8}$ |
| $D_9\wr C_2^2.C_3$ | 9B1 | $9$ | $24$ | not computed | 9B2 | 3C | $e^{2}f^{4}g^{7}h^{8}$ |
| $D_9\wr C_2^2.C_3$ | 9B2 | $9$ | $24$ | not computed | 9B4 | 3C | $e^{4}f^{8}g^{5}h^{7}$ |
| $D_9\wr C_2^2.C_3$ | 9B4 | $9$ | $24$ | not computed | 9B1 | 3C | $e^{8}f^{16}gh^{5}$ |
| $D_9\wr C_2^2.C_3$ | 9C1 | $9$ | $32$ | not computed | 9C2 | 3D | $ef^{12}g^{2}h$ |
| $D_9\wr C_2^2.C_3$ | 9C2 | $9$ | $32$ | not computed | 9C4 | 3D | $e^{2}f^{6}g^{4}h^{2}$ |
| $D_9\wr C_2^2.C_3$ | 9C4 | $9$ | $32$ | not computed | 9C1 | 3D | $e^{4}f^{12}g^{8}h^{4}$ |
| $D_9\wr C_2^2.C_3$ | 9D | $9$ | $48$ | not computed | 9D | 3B | $f^{8}h^{2}$ |
| $D_9\wr C_2^2.C_3$ | 9E1 | $9$ | $48$ | not computed | 9E2 | 3C | $e^{5}f^{4}g^{7}h^{8}$ |
| $D_9\wr C_2^2.C_3$ | 9E2 | $9$ | $48$ | not computed | 9E4 | 3C | $ef^{8}g^{5}h^{7}$ |
| $D_9\wr C_2^2.C_3$ | 9E4 | $9$ | $48$ | not computed | 9E1 | 3C | $e^{2}f^{16}gh^{5}$ |
| $D_9\wr C_2^2.C_3$ | 9F1 | $9$ | $48$ | not computed | 9F2 | 3A | $e^{4}h^{6}$ |
| $D_9\wr C_2^2.C_3$ | 9F2 | $9$ | $48$ | not computed | 9F4 | 3A | $e^{8}h^{3}$ |
| $D_9\wr C_2^2.C_3$ | 9F4 | $9$ | $48$ | not computed | 9F1 | 3A | $e^{7}h^{6}$ |
| $D_9\wr C_2^2.C_3$ | 9G1 | $9$ | $64$ | not computed | 9G2 | 3D | $e^{8}f^{8}g^{6}h^{5}$ |
| $D_9\wr C_2^2.C_3$ | 9G2 | $9$ | $64$ | not computed | 9G4 | 3D | $e^{7}f^{16}g^{3}h$ |
| $D_9\wr C_2^2.C_3$ | 9G4 | $9$ | $64$ | not computed | 9G1 | 3D | $e^{5}f^{14}g^{6}h^{2}$ |
| $D_9\wr C_2^2.C_3$ | 9H | $9$ | $96$ | not computed | 9H | 3D | $e^{4}f^{4}g^{7}h^{4}$ |
| $D_9\wr C_2^2.C_3$ | 9I | $9$ | $96$ | not computed | 9I | 3D | $e^{4}f^{4}g^{7}h$ |
| $D_9\wr C_2^2.C_3$ | 9J1 | $9$ | $96$ | not computed | 9J2 | 3C | $e^{4}f^{4}g^{7}$ |
| $D_9\wr C_2^2.C_3$ | 9J2 | $9$ | $96$ | not computed | 9J4 | 3C | $e^{8}f^{8}g^{5}$ |
| $D_9\wr C_2^2.C_3$ | 9J4 | $9$ | $96$ | not computed | 9J1 | 3C | $e^{7}f^{16}g$ |