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Elements of the group are displayed as permutations of degree 15.

Group Label Order Size Centralizer Powers Representative
2P 3P
$S_3^4:S_3$ 1A $1$ $1$ $S_3^4:S_3$ 1A 1A $()$
$S_3^4:S_3$ 2A $2$ $6$ $C_6\times S_3^3$ 1A 2A $(13,15)$
$S_3^4:S_3$ 2B $2$ $6$ $C_6\times S_3^3$ 1A 2B $(1,2)(3,5)(4,6)$
$S_3^4:S_3$ 2C $2$ $9$ $D_6^2:S_3$ 1A 2C $(12,14)(13,15)$
$S_3^4:S_3$ 2D $2$ $9$ $D_6^2:S_3$ 1A 2D $(3,4)(5,6)$
$S_3^4:S_3$ 2E $2$ $18$ $C_3\times D_6^2$ 1A 2E $(1,2)(3,5)(4,6)(13,15)$
$S_3^4:S_3$ 2F $2$ $18$ $C_3\times D_6^2$ 1A 2F $(1,2)(3,5)(4,6)(12,14)$
$S_3^4:S_3$ 2G $2$ $54$ $C_6^2:C_2^2$ 1A 2G $(3,4)(5,6)(13,15)$
$S_3^4:S_3$ 2H $2$ $54$ $C_6^2:C_2^2$ 1A 2H $(1,2)(3,5)(4,6)(12,14)(13,15)$
$S_3^4:S_3$ 2I $2$ $81$ $C_2^4:S_3$ 1A 2I $(1,3)(5,6)(10,15)(11,14)$
$S_3^4:S_3$ 2J $2$ $108$ $S_3\times D_6$ 1A 2J $(2,5)(8,9)(10,14)(11,15)(12,13)$
$S_3^4:S_3$ 3A $3$ $2$ $C_3\times S_3^4$ 3A 1A $(7,9,8)$
$S_3^4:S_3$ 3B $3$ $4$ $C_3^2\times S_3^3$ 3B 1A $(11,14,12)$
$S_3^4:S_3$ 3C $3$ $4$ $C_3^2\times S_3^3$ 3C 1A $(1,3,4)(2,5,6)$
$S_3^4:S_3$ 3D $3$ $4$ $C_3^5:D_4$ 3D 1A $(2,6,5)$
$S_3^4:S_3$ 3E $3$ $4$ $C_3^5:D_4$ 3E 1A $(10,15,13)(11,14,12)$
$S_3^4:S_3$ 3F1 $3$ $4$ $C_3^2\times S_3^3$ 3F-1 1A $(7,9,8)(11,14,12)$
$S_3^4:S_3$ 3F-1 $3$ $4$ $C_3^2\times S_3^3$ 3F1 1A $(7,8,9)(11,12,14)$
$S_3^4:S_3$ 3G1 $3$ $4$ $C_3^2\times S_3^3$ 3G-1 1A $(1,3,4)(2,5,6)(7,9,8)$
$S_3^4:S_3$ 3G-1 $3$ $4$ $C_3^2\times S_3^3$ 3G1 1A $(1,4,3)(2,6,5)(7,8,9)$
$S_3^4:S_3$ 3H $3$ $8$ $C_3^3\times S_3^2$ 3H 1A $(1,3,4)(2,5,6)(11,14,12)$
$S_3^4:S_3$ 3I $3$ $8$ $C_3^3\times S_3^2$ 3I 1A $(1,3,4)(2,5,6)(10,15,13)$
$S_3^4:S_3$ 3J $3$ $8$ $C_3^3\times S_3^2$ 3J 1A $(2,6,5)(7,9,8)$
$S_3^4:S_3$ 3K $3$ $8$ $C_3^3\times S_3^2$ 3K 1A $(7,9,8)(10,15,13)(11,14,12)$
$S_3^4:S_3$ 3L1 $3$ $8$ $C_3^3\times S_3^2$ 3L-1 1A $(1,3,4)(2,5,6)(7,9,8)(11,14,12)$
$S_3^4:S_3$ 3L-1 $3$ $8$ $C_3^3\times S_3^2$ 3L1 1A $(1,4,3)(2,6,5)(7,8,9)(11,12,14)$
$S_3^4:S_3$ 3M1 $3$ $8$ $C_3^3\times S_3^2$ 3M-1 1A $(1,3,4)(2,5,6)(7,9,8)(10,15,13)$
$S_3^4:S_3$ 3M-1 $3$ $8$ $C_3^3\times S_3^2$ 3M1 1A $(1,4,3)(2,6,5)(7,8,9)(10,13,15)$
$S_3^4:S_3$ 3N $3$ $16$ $S_3\times C_3^4$ 3N 1A $(2,6,5)(11,14,12)$
$S_3^4:S_3$ 3O $3$ $16$ $S_3\times C_3^4$ 3O 1A $(1,3,4)(2,5,6)(10,15,13)(11,14,12)$
$S_3^4:S_3$ 3P $3$ $16$ $C_3^4:C_6$ 3P 1A $(1,3,4)(10,13,15)(11,14,12)$
$S_3^4:S_3$ 3Q1 $3$ $16$ $S_3\times C_3^4$ 3Q-1 1A $(2,6,5)(7,9,8)(11,14,12)$
$S_3^4:S_3$ 3Q-1 $3$ $16$ $S_3\times C_3^4$ 3Q1 1A $(2,5,6)(7,8,9)(11,12,14)$
$S_3^4:S_3$ 3R1 $3$ $16$ $S_3\times C_3^4$ 3R-1 1A $(1,3,4)(2,5,6)(7,9,8)(10,15,13)(11,14,12)$
$S_3^4:S_3$ 3R-1 $3$ $16$ $S_3\times C_3^4$ 3R1 1A $(1,4,3)(2,6,5)(7,8,9)(10,13,15)(11,12,14)$
$S_3^4:S_3$ 3S $3$ $32$ $C_3^5$ 3S 1A $(2,5,6)(7,8,9)(10,13,15)(11,12,14)$
$S_3^4:S_3$ 4A $4$ $324$ $C_4\times S_3$ 2C 4A $(2,5)(7,9)(10,14,13,11)(12,15)$
$S_3^4:S_3$ 4B $4$ $324$ $C_4\times S_3$ 2D 4B $(1,5)(2,3,6,4)(8,9)(10,11)(12,15)(13,14)$
$S_3^4:S_3$ 4C $4$ $972$ $C_2\times C_4$ 2I 4C $(1,6,3,5)(2,4)(8,9)(10,14,15,11)(12,13)$
$S_3^4:S_3$ 6A1 $6$ $6$ $C_6\times S_3^3$ 3A 2A $(7,8,9)(13,15)$
$S_3^4:S_3$ 6A-1 $6$ $6$ $C_6\times S_3^3$ 3A 2A $(7,8,9)(12,14)$
$S_3^4:S_3$ 6B1 $6$ $6$ $C_6\times S_3^3$ 3A 2B $(1,2)(3,5)(4,6)(7,8,9)$
$S_3^4:S_3$ 6B-1 $6$ $6$ $C_6\times S_3^3$ 3A 2B $(1,2)(3,5)(4,6)(7,9,8)$
$S_3^4:S_3$ 6C $6$ $12$ $C_3^4:C_2^3$ 3B 2A $(11,12,14)(13,15)$
$S_3^4:S_3$ 6D $6$ $12$ $C_3^4:C_2^3$ 3B 2B $(1,2)(3,5)(4,6)(11,12,14)$
$S_3^4:S_3$ 6E $6$ $12$ $C_3^4:C_2^3$ 3B 2B $(1,2)(3,5)(4,6)(10,13,15)$
$S_3^4:S_3$ 6F $6$ $12$ $C_3^4:C_2^3$ 3C 2B $(1,2,3,5,4,6)$
$S_3^4:S_3$ 6G $6$ $12$ $C_3^4:C_2^3$ 3C 2A $(1,3,4)(2,5,6)(13,15)$
$S_3^4:S_3$ 6H $6$ $12$ $C_3^4:C_2^3$ 3C 2A $(1,3,4)(2,5,6)(12,14)$
$S_3^4:S_3$ 6I1 $6$ $12$ $C_3^4:C_2^3$ 3F1 2A $(7,8,9)(11,12,14)(13,15)$
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