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Elements of the group are displayed as permutations of degree 18.
| Group | Label | Order | Size | Centralizer | Powers | Representative | |
|---|---|---|---|---|---|---|---|
| 2P | 3P | ||||||
| $C_3^4:S_3^2$ | 1A | $1$ | $1$ | $C_3^4:S_3^2$ | 1A | 1A | $()$ |
| $C_3^4:S_3^2$ | 2A | $2$ | $27$ | $C_3^2:D_6$ | 1A | 2A | $(1,11)(2,12)(3,10)(4,15)(5,13)(6,14)(7,17)(8,18)(9,16)$ |
| $C_3^4:S_3^2$ | 2B | $2$ | $27$ | $C_3^2:D_6$ | 1A | 2B | $(1,12)(2,11)(3,10)(4,14)(5,13)(6,15)(7,18)(8,17)(9,16)$ |
| $C_3^4:S_3^2$ | 2C | $2$ | $81$ | $C_6\times S_3$ | 1A | 2C | $(2,3)(4,5)(7,8)(11,12)(14,15)(17,18)$ |
| $C_3^4:S_3^2$ | 3A | $3$ | $2$ | $\He_3\wr C_2$ | 3A | 1A | $(1,2,3)(4,6,5)(7,8,9)(10,11,12)(13,15,14)(16,17,18)$ |
| $C_3^4:S_3^2$ | 3B | $3$ | $2$ | $\He_3\wr C_2$ | 3B | 1A | $(1,2,3)(4,6,5)(7,8,9)(10,12,11)(13,14,15)(16,18,17)$ |
| $C_3^4:S_3^2$ | 3C | $3$ | $4$ | $\He_3^2$ | 3C | 1A | $(10,11,12)(13,15,14)(16,17,18)$ |
| $C_3^4:S_3^2$ | 3D1 | $3$ | $6$ | $C_3^4:C_6$ | 3D-1 | 1A | $(1,9,6)(2,7,5)(3,8,4)$ |
| $C_3^4:S_3^2$ | 3D-1 | $3$ | $6$ | $C_3^4:C_6$ | 3D1 | 1A | $(1,6,9)(2,5,7)(3,4,8)$ |
| $C_3^4:S_3^2$ | 3E1 | $3$ | $9$ | $C_3\wr C_2^2$ | 3E-1 | 1A | $(1,8,5)(2,9,4)(3,7,6)(10,18,15)(11,16,14)(12,17,13)$ |
| $C_3^4:S_3^2$ | 3E-1 | $3$ | $9$ | $C_3\wr C_2^2$ | 3E1 | 1A | $(1,5,8)(2,4,9)(3,6,7)(10,15,18)(11,14,16)(12,13,17)$ |
| $C_3^4:S_3^2$ | 3F | $3$ | $12$ | $C_3^2\times \He_3$ | 3F | 1A | $(13,14,15)(16,17,18)$ |
| $C_3^4:S_3^2$ | 3G | $3$ | $12$ | $C_3^2\times \He_3$ | 3G | 1A | $(4,5,6)(7,8,9)(10,11,12)(13,15,14)(16,17,18)$ |
| $C_3^4:S_3^2$ | 3H | $3$ | $12$ | $C_3^2\times \He_3$ | 3H | 1A | $(4,5,6)(7,8,9)(10,12,11)(13,14,15)(16,18,17)$ |
| $C_3^4:S_3^2$ | 3I1 | $3$ | $12$ | $C_3^2\times \He_3$ | 3I-1 | 1A | $(10,13,17)(11,15,18)(12,14,16)$ |
| $C_3^4:S_3^2$ | 3I-1 | $3$ | $12$ | $C_3^2\times \He_3$ | 3I1 | 1A | $(10,16,14)(11,17,13)(12,18,15)$ |
| $C_3^4:S_3^2$ | 3J1 | $3$ | $12$ | $C_3^2\times \He_3$ | 3J-1 | 1A | $(1,2,3)(4,6,5)(7,8,9)(10,13,16)(11,15,17)(12,14,18)$ |
| $C_3^4:S_3^2$ | 3J-1 | $3$ | $12$ | $C_3^2\times \He_3$ | 3J1 | 1A | $(1,2,3)(4,6,5)(7,8,9)(10,16,13)(11,17,15)(12,18,14)$ |
| $C_3^4:S_3^2$ | 3K1 | $3$ | $12$ | $C_3^2\times \He_3$ | 3K-1 | 1A | $(1,2,3)(4,6,5)(7,8,9)(10,13,17)(11,15,18)(12,14,16)$ |
| $C_3^4:S_3^2$ | 3K-1 | $3$ | $12$ | $C_3^2\times \He_3$ | 3K1 | 1A | $(1,2,3)(4,6,5)(7,8,9)(10,16,14)(11,17,13)(12,18,15)$ |
| $C_3^4:S_3^2$ | 3L1 | $3$ | $12$ | $C_3^2\times \He_3$ | 3L-1 | 1A | $(1,2,3)(4,6,5)(7,8,9)(10,13,18)(11,15,16)(12,14,17)$ |
| $C_3^4:S_3^2$ | 3L-1 | $3$ | $12$ | $C_3^2\times \He_3$ | 3L1 | 1A | $(1,2,3)(4,6,5)(7,8,9)(10,16,15)(11,17,14)(12,18,13)$ |
| $C_3^4:S_3^2$ | 3M | $3$ | $18$ | $C_3^2\wr C_2$ | 3M | 1A | $(1,9,6)(2,7,5)(3,8,4)(10,13,16)(11,15,17)(12,14,18)$ |
| $C_3^4:S_3^2$ | 3N | $3$ | $18$ | $C_3^2\wr C_2$ | 3N | 1A | $(1,2,3)(7,9,8)(10,12,11)(16,17,18)$ |
| $C_3^4:S_3^2$ | 3O | $3$ | $18$ | $C_3^2\wr C_2$ | 3O | 1A | $(1,2,3)(7,9,8)(13,14,15)(16,17,18)$ |
| $C_3^4:S_3^2$ | 3P1 | $3$ | $18$ | $C_3^2\wr C_2$ | 3P-1 | 1A | $(1,8,6)(2,9,5)(3,7,4)(10,18,13)(11,16,15)(12,17,14)$ |
| $C_3^4:S_3^2$ | 3P-1 | $3$ | $18$ | $C_3^2\wr C_2$ | 3P1 | 1A | $(1,6,8)(2,5,9)(3,4,7)(10,13,18)(11,15,16)(12,14,17)$ |
| $C_3^4:S_3^2$ | 3Q1 | $3$ | $18$ | $C_3^2\wr C_2$ | 3Q-1 | 1A | $(1,8,4)(2,9,6)(3,7,5)(10,16,14)(11,17,13)(12,18,15)$ |
| $C_3^4:S_3^2$ | 3Q-1 | $3$ | $18$ | $C_3^2\wr C_2$ | 3Q1 | 1A | $(1,4,8)(2,6,9)(3,5,7)(10,14,16)(11,13,17)(12,15,18)$ |
| $C_3^4:S_3^2$ | 3R | $3$ | $36$ | $C_3^4$ | 3R | 1A | $(1,4,8)(2,6,9)(3,5,7)(10,16,14)(11,17,13)(12,18,15)$ |
| $C_3^4:S_3^2$ | 3S | $3$ | $36$ | $C_3^4$ | 3S | 1A | $(1,4,8)(2,6,9)(3,5,7)(10,16,15)(11,17,14)(12,18,13)$ |
| $C_3^4:S_3^2$ | 3T1 | $3$ | $36$ | $C_3^4$ | 3T-1 | 1A | $(4,5,6)(7,8,9)(10,13,16)(11,15,17)(12,14,18)$ |
| $C_3^4:S_3^2$ | 3T-1 | $3$ | $36$ | $C_3^4$ | 3T1 | 1A | $(4,5,6)(7,8,9)(10,16,13)(11,17,15)(12,18,14)$ |
| $C_3^4:S_3^2$ | 3U1 | $3$ | $36$ | $C_3^4$ | 3U-1 | 1A | $(4,5,6)(7,8,9)(10,13,17)(11,15,18)(12,14,16)$ |
| $C_3^4:S_3^2$ | 3U-1 | $3$ | $36$ | $C_3^4$ | 3U1 | 1A | $(4,5,6)(7,8,9)(10,16,14)(11,17,13)(12,18,15)$ |
| $C_3^4:S_3^2$ | 3V1 | $3$ | $36$ | $C_3^4$ | 3V-1 | 1A | $(4,5,6)(7,8,9)(10,13,18)(11,15,16)(12,14,17)$ |
| $C_3^4:S_3^2$ | 3V-1 | $3$ | $36$ | $C_3^4$ | 3V1 | 1A | $(4,5,6)(7,8,9)(10,16,15)(11,17,14)(12,18,13)$ |
| $C_3^4:S_3^2$ | 3W1 | $3$ | $36$ | $C_3^4$ | 3W-1 | 1A | $(1,4,7)(2,6,8)(3,5,9)(10,13,17)(11,15,18)(12,14,16)$ |
| $C_3^4:S_3^2$ | 3W-1 | $3$ | $36$ | $C_3^4$ | 3W1 | 1A | $(1,7,4)(2,8,6)(3,9,5)(10,16,14)(11,17,13)(12,18,15)$ |
| $C_3^4:S_3^2$ | 3X1 | $3$ | $36$ | $C_3^4$ | 3X-1 | 1A | $(1,4,7)(2,6,8)(3,5,9)(10,16,14)(11,17,13)(12,18,15)$ |
| $C_3^4:S_3^2$ | 3X-1 | $3$ | $36$ | $C_3^4$ | 3X1 | 1A | $(1,4,8)(2,6,9)(3,5,7)(10,16,13)(11,17,15)(12,18,14)$ |
| $C_3^4:S_3^2$ | 6A | $6$ | $54$ | $C_2\times \He_3$ | 3A | 2A | $(1,10,2,11,3,12)(4,13,6,15,5,14)(7,16,8,17,9,18)$ |
| $C_3^4:S_3^2$ | 6B | $6$ | $54$ | $C_2\times \He_3$ | 3B | 2B | $(1,10,2,12,3,11)(4,13,6,14,5,15)(7,16,8,18,9,17)$ |
| $C_3^4:S_3^2$ | 6C1 | $6$ | $81$ | $C_6\times S_3$ | 3E1 | 2A | $(1,12,8,17,5,13)(2,10,9,18,4,15)(3,11,7,16,6,14)$ |
| $C_3^4:S_3^2$ | 6C-1 | $6$ | $81$ | $C_6\times S_3$ | 3E-1 | 2A | $(1,13,5,17,8,12)(2,15,4,18,9,10)(3,14,6,16,7,11)$ |
| $C_3^4:S_3^2$ | 6D1 | $6$ | $81$ | $C_6\times S_3$ | 3E-1 | 2C | $(1,9,6)(2,8,5,3,7,4)(10,16,13)(11,18,15,12,17,14)$ |
| $C_3^4:S_3^2$ | 6D-1 | $6$ | $81$ | $C_6\times S_3$ | 3E1 | 2C | $(1,6,9)(2,4,7,3,5,8)(10,13,16)(11,14,17,12,15,18)$ |
| $C_3^4:S_3^2$ | 6E1 | $6$ | $81$ | $C_6\times S_3$ | 3E1 | 2B | $(1,15,9,12,6,16)(2,13,7,11,5,18)(3,14,8,10,4,17)$ |
| $C_3^4:S_3^2$ | 6E-1 | $6$ | $81$ | $C_6\times S_3$ | 3E-1 | 2B | $(1,16,6,12,9,15)(2,18,5,11,7,13)(3,17,4,10,8,14)$ |
| $C_3^4:S_3^2$ | 6F | $6$ | $162$ | $C_3\times C_6$ | 3M | 2C | $(1,6,9)(2,4,7,3,5,8)(10,16,13)(11,18,15,12,17,14)$ |