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Elements of the group are displayed as permutations of degree 18.
| Group | Label | Order | Size | Centralizer | Powers | Representative | |
|---|---|---|---|---|---|---|---|
| 2P | 3P | ||||||
| $C_6^3:S_3^2$ | 1A | $1$ | $1$ | $C_6^3:S_3^2$ | 1A | 1A | $()$ |
| $C_6^3:S_3^2$ | 2A | $2$ | $1$ | $C_6^3:S_3^2$ | 1A | 2A | $(8,9)$ |
| $C_6^3:S_3^2$ | 2B | $2$ | $3$ | $(C_3^3\times C_6).C_2^4$ | 1A | 2B | $(1,2)(3,4)(8,9)$ |
| $C_6^3:S_3^2$ | 2C | $2$ | $3$ | $(C_3^3\times C_6).C_2^4$ | 1A | 2C | $(1,2)(3,4)$ |
| $C_6^3:S_3^2$ | 2D | $2$ | $3$ | $C_6^3:D_6$ | 1A | 2D | $(6,7)$ |
| $C_6^3:S_3^2$ | 2E | $2$ | $3$ | $C_6^3:D_6$ | 1A | 2E | $(6,7)(8,9)$ |
| $C_6^3:S_3^2$ | 2F | $2$ | $9$ | $C_6^3:C_2^2$ | 1A | 2F | $(1,2)(3,4)(6,7)$ |
| $C_6^3:S_3^2$ | 2G | $2$ | $9$ | $C_6^3:C_2^2$ | 1A | 2G | $(1,2)(3,4)(6,7)(8,9)$ |
| $C_6^3:S_3^2$ | 2H | $2$ | $54$ | $C_6^2:C_2^2$ | 1A | 2H | $(3,4)(10,16)(12,17)(13,14)(15,18)$ |
| $C_6^3:S_3^2$ | 2I | $2$ | $54$ | $C_6^2:C_2^2$ | 1A | 2I | $(3,4)(8,9)(10,16)(12,17)(13,14)(15,18)$ |
| $C_6^3:S_3^2$ | 2J | $2$ | $162$ | $C_2^3\times C_6$ | 1A | 2J | $(3,4)(6,7)(8,9)(10,12)(11,18)(13,16)(14,17)$ |
| $C_6^3:S_3^2$ | 2K | $2$ | $162$ | $C_2^3\times C_6$ | 1A | 2K | $(1,3)(6,7)(10,18)(11,13)(12,14)(15,17)$ |
| $C_6^3:S_3^2$ | 3A | $3$ | $2$ | $C_3\times C_6^3:S_3$ | 3A | 1A | $(5,7,6)$ |
| $C_6^3:S_3^2$ | 3B | $3$ | $2$ | $C_6\wr C_3\times S_3$ | 3B | 1A | $(10,17,13)(11,15,18)(12,16,14)$ |
| $C_6^3:S_3^2$ | 3C1 | $3$ | $3$ | $(C_3^3\times C_6).C_2^4$ | 3C-1 | 1A | $(11,18,15)(12,16,14)$ |
| $C_6^3:S_3^2$ | 3C-1 | $3$ | $3$ | $(C_3^3\times C_6).C_2^4$ | 3C1 | 1A | $(11,15,18)(12,14,16)$ |
| $C_6^3:S_3^2$ | 3D | $3$ | $4$ | $C_6^3:C_3^2$ | 3D | 1A | $(5,7,6)(10,17,13)(11,15,18)(12,16,14)$ |
| $C_6^3:S_3^2$ | 3E | $3$ | $6$ | $S_3\times C_6^3$ | 3E | 1A | $(12,16,14)$ |
| $C_6^3:S_3^2$ | 3F | $3$ | $6$ | $S_3\times C_6^3$ | 3F | 1A | $(11,18,15)(12,14,16)$ |
| $C_6^3:S_3^2$ | 3G | $3$ | $6$ | $S_3\times C_6^3$ | 3G | 1A | $(10,17,13)(11,18,15)(12,16,14)$ |
| $C_6^3:S_3^2$ | 3H1 | $3$ | $6$ | $C_6^3:C_6$ | 3H-1 | 1A | $(5,7,6)(11,18,15)(12,16,14)$ |
| $C_6^3:S_3^2$ | 3H-1 | $3$ | $6$ | $C_6^3:C_6$ | 3H1 | 1A | $(5,6,7)(11,15,18)(12,14,16)$ |
| $C_6^3:S_3^2$ | 3I | $3$ | $12$ | $C_3\times C_6^3$ | 3I | 1A | $(5,7,6)(12,16,14)$ |
| $C_6^3:S_3^2$ | 3J | $3$ | $12$ | $C_3\times C_6^3$ | 3J | 1A | $(5,7,6)(11,18,15)(12,14,16)$ |
| $C_6^3:S_3^2$ | 3K | $3$ | $12$ | $C_3\times C_6^3$ | 3K | 1A | $(5,7,6)(10,17,13)(11,18,15)(12,16,14)$ |
| $C_6^3:S_3^2$ | 3L | $3$ | $72$ | $C_3^2\times D_6$ | 3L | 1A | $(2,4,3)(10,12,11)(13,14,18)(15,17,16)$ |
| $C_6^3:S_3^2$ | 3M | $3$ | $144$ | $C_3^2\times C_6$ | 3M | 1A | $(1,4,3)(5,7,6)(10,18,14)(11,12,17)(13,15,16)$ |
| $C_6^3:S_3^2$ | 4A | $4$ | $54$ | $C_{12}\times D_6$ | 2C | 4A | $(1,4,3,2)(10,15)(11,17)(13,18)(14,16)$ |
| $C_6^3:S_3^2$ | 4B | $4$ | $54$ | $C_{12}\times D_6$ | 2C | 4B | $(1,4,3,2)(8,9)(10,15)(11,17)(13,18)(14,16)$ |
| $C_6^3:S_3^2$ | 4C | $4$ | $162$ | $C_2^2\times C_{12}$ | 2C | 4C | $(1,2,4,3)(6,7)(10,17)(11,12)(14,15)(16,18)$ |
| $C_6^3:S_3^2$ | 4D | $4$ | $162$ | $C_2^2\times C_{12}$ | 2C | 4D | $(1,4,2,3)(5,6)(8,9)(10,14)(11,18)(12,13)(16,17)$ |
| $C_6^3:S_3^2$ | 6A | $6$ | $2$ | $C_3\times C_6^3:S_3$ | 3A | 2A | $(5,6,7)(8,9)$ |
| $C_6^3:S_3^2$ | 6B | $6$ | $2$ | $C_6\wr C_3\times S_3$ | 3B | 2A | $(8,9)(10,13,17)(11,18,15)(12,14,16)$ |
| $C_6^3:S_3^2$ | 6C1 | $6$ | $3$ | $(C_3^3\times C_6).C_2^4$ | 3C1 | 2A | $(8,9)(11,15,18)(12,14,16)$ |
| $C_6^3:S_3^2$ | 6C-1 | $6$ | $3$ | $(C_3^3\times C_6).C_2^4$ | 3C-1 | 2A | $(8,9)(11,18,15)(12,16,14)$ |
| $C_6^3:S_3^2$ | 6D1 | $6$ | $3$ | $(C_3^3\times C_6).C_2^4$ | 3C1 | 2B | $(1,2)(3,4)(8,9)(10,13,17)(12,16,14)$ |
| $C_6^3:S_3^2$ | 6D-1 | $6$ | $3$ | $(C_3^3\times C_6).C_2^4$ | 3C-1 | 2B | $(1,2)(3,4)(8,9)(10,17,13)(12,14,16)$ |
| $C_6^3:S_3^2$ | 6E1 | $6$ | $3$ | $(C_3^3\times C_6).C_2^4$ | 3C-1 | 2C | $(1,3)(2,4)(10,13,17)(11,15,18)$ |
| $C_6^3:S_3^2$ | 6E-1 | $6$ | $3$ | $(C_3^3\times C_6).C_2^4$ | 3C1 | 2C | $(1,3)(2,4)(10,17,13)(11,18,15)$ |
| $C_6^3:S_3^2$ | 6F | $6$ | $4$ | $C_6^3:C_3^2$ | 3D | 2A | $(5,6,7)(8,9)(10,13,17)(11,18,15)(12,14,16)$ |
| $C_6^3:S_3^2$ | 6G | $6$ | $6$ | $S_3\times C_6^3$ | 3E | 2A | $(8,9)(12,14,16)$ |
| $C_6^3:S_3^2$ | 6H | $6$ | $6$ | $S_3\times C_6^3$ | 3F | 2A | $(8,9)(11,15,18)(12,16,14)$ |
| $C_6^3:S_3^2$ | 6I | $6$ | $6$ | $S_3\times C_6^3$ | 3G | 2A | $(8,9)(10,13,17)(11,15,18)(12,14,16)$ |
| $C_6^3:S_3^2$ | 6J | $6$ | $6$ | $S_3\times C_6^3$ | 3E | 2C | $(1,2)(3,4)(11,15,18)$ |
| $C_6^3:S_3^2$ | 6K | $6$ | $6$ | $S_3\times C_6^3$ | 3F | 2C | $(1,2)(3,4)(10,13,17)(12,14,16)$ |
| $C_6^3:S_3^2$ | 6L | $6$ | $6$ | $S_3\times C_6^3$ | 3G | 2C | $(1,2)(3,4)(10,13,17)(11,15,18)(12,14,16)$ |
| $C_6^3:S_3^2$ | 6M | $6$ | $6$ | $S_3\times C_6^3$ | 3B | 2C | $(1,2)(3,4)(10,13,17)(11,18,15)(12,14,16)$ |
| $C_6^3:S_3^2$ | 6N | $6$ | $6$ | $S_3\times C_6^3$ | 3E | 2B | $(1,2)(3,4)(8,9)(11,15,18)$ |
| $C_6^3:S_3^2$ | 6O | $6$ | $6$ | $S_3\times C_6^3$ | 3F | 2B | $(1,2)(3,4)(8,9)(10,13,17)(12,14,16)$ |
| $C_6^3:S_3^2$ | 6P | $6$ | $6$ | $S_3\times C_6^3$ | 3G | 2B | $(1,2)(3,4)(8,9)(10,13,17)(11,15,18)(12,14,16)$ |