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Elements of the group are displayed as permutations of degree 24.

Group Label Order Size Centralizer Powers Representative
2P 3P
$(C_3^3\times \He_3):D_{12}$ 1A $1$ $1$ $(C_3^3\times \He_3):D_{12}$ 1A 1A $()$
$(C_3^3\times \He_3):D_{12}$ 2A $2$ $9$ $(C_3\times \He_3):D_{12}$ 1A 2A $(21,23)(22,24)$
$(C_3^3\times \He_3):D_{12}$ 2B $2$ $162$ $C_3^2\times D_6$ 1A 2B $(1,7)(2,18)(3,13)(4,6)(5,17)(8,10)(9,14)(11,16)(12,15)(19,22)(20,23)(21,24)$
$(C_3^3\times \He_3):D_{12}$ 2C $2$ $486$ $C_6\times S_3$ 1A 2C $(3,17)(5,13)(6,15)(7,16)(9,12)(11,18)(20,22)$
$(C_3^3\times \He_3):D_{12}$ 3A $3$ $2$ $C_3^6:D_6$ 3A 1A $(1,18,11)(2,16,7)(3,8,17)(4,12,9)(5,13,10)(6,15,14)$
$(C_3^3\times \He_3):D_{12}$ 3B $3$ $2$ $(C_3^3\times \He_3):C_{12}$ 3B 1A $(1,11,18)(2,16,7)(3,8,17)(4,9,12)(5,10,13)(6,15,14)$
$(C_3^3\times \He_3):D_{12}$ 3C $3$ $4$ $C_3^6.C_6$ 3C 1A $(2,16,7)(3,8,17)(6,15,14)$
$(C_3^3\times \He_3):D_{12}$ 3D $3$ $4$ $C_3^3.\He_3.C_6$ 3D 1A $(19,23,21)(20,24,22)$
$(C_3^3\times \He_3):D_{12}$ 3E $3$ $4$ $C_3^6.C_6$ 3E 1A $(19,23,21)$
$(C_3^3\times \He_3):D_{12}$ 3F1 $3$ $4$ $C_3^3.\He_3.C_6$ 3F-1 1A $(1,11,18)(2,7,16)(3,17,8)(4,9,12)(5,10,13)(6,14,15)(19,21,23)(20,22,24)$
$(C_3^3\times \He_3):D_{12}$ 3F-1 $3$ $4$ $C_3^3.\He_3.C_6$ 3F1 1A $(1,18,11)(2,16,7)(3,8,17)(4,12,9)(5,13,10)(6,15,14)(19,23,21)(20,24,22)$
$(C_3^3\times \He_3):D_{12}$ 3G $3$ $6$ $C_3^4:C_6^2$ 3G 1A $(3,17,8)(4,12,9)(5,10,13)(6,15,14)$
$(C_3^3\times \He_3):D_{12}$ 3H $3$ $6$ $C_3^4:C_6^2$ 3H 1A $(2,16,7)(4,12,9)(5,10,13)(6,14,15)$
$(C_3^3\times \He_3):D_{12}$ 3I $3$ $6$ $C_3^4:C_6^2$ 3I 1A $(2,16,7)(3,17,8)(4,9,12)(5,13,10)$
$(C_3^3\times \He_3):D_{12}$ 3J $3$ $6$ $C_3^5:C_{12}$ 3J 1A $(1,11,18)(3,17,8)(4,12,9)(6,15,14)$
$(C_3^3\times \He_3):D_{12}$ 3K $3$ $6$ $C_3^5:C_{12}$ 3K 1A $(1,11,18)(2,16,7)(4,12,9)(6,14,15)$
$(C_3^3\times \He_3):D_{12}$ 3L $3$ $6$ $C_3^5:C_{12}$ 3L 1A $(1,18,11)(2,16,7)(3,17,8)(4,9,12)$
$(C_3^3\times \He_3):D_{12}$ 3M $3$ $8$ $C_3^2\times C_3^4:C_3$ 3M 1A $(2,7,16)(3,17,8)(6,14,15)(20,22,24)$
$(C_3^3\times \He_3):D_{12}$ 3N $3$ $8$ $C_3^2\times C_3^4:C_3$ 3N 1A $(2,7,16)(3,17,8)(6,14,15)(19,21,23)$
$(C_3^3\times \He_3):D_{12}$ 3O $3$ $8$ $C_3^2\times C_3^4:C_3$ 3O 1A $(1,11,18)(2,7,16)(3,17,8)(4,9,12)(5,10,13)(6,14,15)(20,22,24)$
$(C_3^3\times \He_3):D_{12}$ 3P $3$ $8$ $C_3^2\times C_3^4:C_3$ 3P 1A $(1,11,18)(2,16,7)(3,8,17)(4,9,12)(5,10,13)(6,15,14)(20,22,24)$
$(C_3^3\times \He_3):D_{12}$ 3Q $3$ $8$ $C_3^2\times C_3^4:C_3$ 3Q 1A $(1,11,18)(2,16,7)(3,8,17)(4,9,12)(5,10,13)(6,15,14)(19,21,23)(20,22,24)$
$(C_3^3\times \He_3):D_{12}$ 3R1 $3$ $8$ $C_3^2\times C_3^4:C_3$ 3R-1 1A $(2,7,16)(3,17,8)(6,14,15)(19,21,23)(20,22,24)$
$(C_3^3\times \He_3):D_{12}$ 3R-1 $3$ $8$ $C_3^2\times C_3^4:C_3$ 3R1 1A $(2,7,16)(3,17,8)(6,14,15)(19,21,23)(20,24,22)$
$(C_3^3\times \He_3):D_{12}$ 3S $3$ $12$ $C_3^5:C_6$ 3S 1A $(4,12,9)(5,10,13)$
$(C_3^3\times \He_3):D_{12}$ 3T $3$ $12$ $C_3^5:C_6$ 3T 1A $(2,16,7)(3,8,17)(4,12,9)(5,10,13)(6,15,14)$
$(C_3^3\times \He_3):D_{12}$ 3U $3$ $12$ $C_3^5:C_6$ 3U 1A $(2,16,7)(3,8,17)(4,9,12)(5,13,10)(6,15,14)$
$(C_3^3\times \He_3):D_{12}$ 3V1 $3$ $12$ $C_3^3\wr C_2$ 3V-1 1A $(3,17,8)(4,12,9)(5,10,13)(6,15,14)(19,23,21)(20,24,22)$
$(C_3^3\times \He_3):D_{12}$ 3V-1 $3$ $12$ $C_3^3\wr C_2$ 3V1 1A $(3,8,17)(4,9,12)(5,13,10)(6,14,15)(19,21,23)(20,22,24)$
$(C_3^3\times \He_3):D_{12}$ 3W1 $3$ $12$ $C_3^3\wr C_2$ 3W-1 1A $(1,11,18)(3,17,8)(5,13,10)(6,15,14)(19,21,23)(20,22,24)$
$(C_3^3\times \He_3):D_{12}$ 3W-1 $3$ $12$ $C_3^3\wr C_2$ 3W1 1A $(1,18,11)(3,8,17)(5,10,13)(6,14,15)(19,23,21)(20,24,22)$
$(C_3^3\times \He_3):D_{12}$ 3X1 $3$ $12$ $C_3^3\wr C_2$ 3X-1 1A $(2,16,7)(4,12,9)(5,10,13)(6,14,15)(19,21,23)(20,22,24)$
$(C_3^3\times \He_3):D_{12}$ 3X-1 $3$ $12$ $C_3^3\wr C_2$ 3X1 1A $(2,7,16)(4,9,12)(5,13,10)(6,15,14)(19,23,21)(20,24,22)$
$(C_3^3\times \He_3):D_{12}$ 3Y $3$ $18$ $C_3^3:S_3^2$ 3Y 1A $(1,13,12)(2,15,17)(3,16,14)(4,11,5)(6,8,7)(9,18,10)$
$(C_3^3\times \He_3):D_{12}$ 3Z $3$ $24$ $C_3^6$ 3Z 1A $(4,9,12)(5,13,10)(20,22,24)$
$(C_3^3\times \He_3):D_{12}$ 3AA $3$ $24$ $C_3^6$ 3AA 1A $(4,9,12)(5,13,10)(19,21,23)$
$(C_3^3\times \He_3):D_{12}$ 3AB $3$ $24$ $C_3^6$ 3AB 1A $(3,8,17)(4,9,12)(5,13,10)(6,14,15)(20,22,24)$
$(C_3^3\times \He_3):D_{12}$ 3AC $3$ $24$ $C_3^6$ 3AC 1A $(3,8,17)(4,12,9)(5,10,13)(6,14,15)(20,22,24)$
$(C_3^3\times \He_3):D_{12}$ 3AD $3$ $24$ $C_3^6$ 3AD 1A $(3,8,17)(4,12,9)(5,10,13)(6,14,15)(19,21,23)(20,22,24)$
$(C_3^3\times \He_3):D_{12}$ 3AE $3$ $24$ $C_3^6$ 3AE 1A $(2,7,16)(4,9,12)(5,13,10)(6,15,14)(20,22,24)$
$(C_3^3\times \He_3):D_{12}$ 3AF $3$ $24$ $C_3^6$ 3AF 1A $(2,7,16)(4,12,9)(5,10,13)(6,15,14)(20,22,24)$
$(C_3^3\times \He_3):D_{12}$ 3AG $3$ $24$ $C_3^6$ 3AG 1A $(2,7,16)(4,12,9)(5,10,13)(6,15,14)(19,21,23)(20,22,24)$
$(C_3^3\times \He_3):D_{12}$ 3AH $3$ $24$ $C_3^6$ 3AH 1A $(2,7,16)(3,8,17)(4,9,12)(5,13,10)(20,22,24)$
$(C_3^3\times \He_3):D_{12}$ 3AI $3$ $24$ $C_3^6$ 3AI 1A $(2,7,16)(3,8,17)(4,9,12)(5,13,10)(19,21,23)(20,22,24)$
$(C_3^3\times \He_3):D_{12}$ 3AJ $3$ $24$ $C_3^6$ 3AJ 1A $(2,7,16)(3,8,17)(4,12,9)(5,10,13)(20,22,24)$
$(C_3^3\times \He_3):D_{12}$ 3AK $3$ $24$ $C_3^6$ 3AK 1A $(2,7,16)(3,17,8)(4,9,12)(5,13,10)(6,14,15)(20,22,24)$
$(C_3^3\times \He_3):D_{12}$ 3AL $3$ $24$ $C_3^6$ 3AL 1A $(2,7,16)(3,17,8)(4,9,12)(5,13,10)(6,14,15)(19,21,23)$
$(C_3^3\times \He_3):D_{12}$ 3AM $3$ $24$ $C_3^6$ 3AM 1A $(2,7,16)(3,17,8)(4,12,9)(5,10,13)(6,14,15)(20,22,24)$
$(C_3^3\times \He_3):D_{12}$ 3AN $3$ $24$ $C_3^6$ 3AN 1A $(2,7,16)(3,17,8)(4,12,9)(5,10,13)(6,14,15)(19,21,23)$
$(C_3^3\times \He_3):D_{12}$ 3AO1 $3$ $24$ $C_3^6$ 3AO-1 1A $(4,9,12)(5,13,10)(19,21,23)(20,22,24)$
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