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Elements of the group are displayed as permutations of degree 12.

Group Label Order Size Centralizer Powers Representative
2P 3P 5P
$S_3^2\times A_6$ 1A $1$ $1$ $S_3^2\times A_6$ 1A 1A 1A $()$
$S_3^2\times A_6$ 2A $2$ $3$ $D_6\times A_6$ 1A 2A 2A $(7,9)(8,10)(11,12)$
$S_3^2\times A_6$ 2B $2$ $3$ $D_6\times A_6$ 1A 2B 2B $(7,9)(8,11)(10,12)$
$S_3^2\times A_6$ 2C $2$ $9$ $C_2^2\times A_6$ 1A 2C 2C $(7,10)(8,12)$
$S_3^2\times A_6$ 2D $2$ $45$ $D_4\times S_3^2$ 1A 2D 2D $(1,6)(4,5)$
$S_3^2\times A_6$ 2E $2$ $135$ $D_4\times D_6$ 1A 2E 2E $(1,3)(2,6)(7,9)(8,11)(10,12)$
$S_3^2\times A_6$ 2F $2$ $135$ $D_4\times D_6$ 1A 2F 2F $(2,5)(4,6)(7,9)(8,10)(11,12)$
$S_3^2\times A_6$ 2G $2$ $405$ $C_2^2\times D_4$ 1A 2G 2G $(1,5)(3,6)(9,12)(10,11)$
$S_3^2\times A_6$ 3A $3$ $2$ $C_3\times S_3\times A_6$ 3A 1A 3A $(7,11,10)(8,12,9)$
$S_3^2\times A_6$ 3B $3$ $2$ $C_3\times S_3\times A_6$ 3B 1A 3B $(7,11,10)(8,9,12)$
$S_3^2\times A_6$ 3C $3$ $4$ $C_3^2\times A_6$ 3C 1A 3C $(7,10,11)$
$S_3^2\times A_6$ 3D $3$ $40$ $C_3^2\times S_3^2$ 3D 1A 3D $(1,2,5)$
$S_3^2\times A_6$ 3E $3$ $40$ $C_3^2\times S_3^2$ 3E 1A 3E $(1,2,5)(3,6,4)$
$S_3^2\times A_6$ 3F $3$ $80$ $S_3\times C_3^3$ 3F 1A 3F $(1,3,2)(4,5,6)(7,10,11)(8,9,12)$
$S_3^2\times A_6$ 3G $3$ $80$ $S_3\times C_3^3$ 3G 1A 3G $(1,6,2)(7,11,10)(8,12,9)$
$S_3^2\times A_6$ 3H $3$ $80$ $S_3\times C_3^3$ 3H 1A 3H $(1,3,6)(2,4,5)(7,10,11)(8,12,9)$
$S_3^2\times A_6$ 3I $3$ $80$ $S_3\times C_3^3$ 3I 1A 3I $(2,4,6)(7,11,10)(8,9,12)$
$S_3^2\times A_6$ 3J $3$ $160$ $C_3^4$ 3J 1A 3J $(1,2,5)(3,4,6)(8,12,9)$
$S_3^2\times A_6$ 3K $3$ $160$ $C_3^4$ 3K 1A 3K $(1,2,4)(8,12,9)$
$S_3^2\times A_6$ 4A $4$ $90$ $C_4\times S_3^2$ 2D 4A 4A $(1,6)(2,3,4,5)$
$S_3^2\times A_6$ 4B $4$ $270$ $C_4\times D_6$ 2D 4B 4B $(1,5,6,4)(2,3)(7,12)(8,10)(9,11)$
$S_3^2\times A_6$ 4C $4$ $270$ $C_4\times D_6$ 2D 4C 4C $(1,5,4,2)(3,6)(7,9)(8,10)(11,12)$
$S_3^2\times A_6$ 4D $4$ $810$ $C_2^2\times C_4$ 2D 4D 4D $(1,3,6,5)(2,4)(7,10)(8,9)$
$S_3^2\times A_6$ 5A1 $5$ $72$ $C_5\times S_3^2$ 5A2 5A2 1A $(1,2,4,6,3)$
$S_3^2\times A_6$ 5A2 $5$ $72$ $C_5\times S_3^2$ 5A1 5A1 1A $(1,4,3,2,6)$
$S_3^2\times A_6$ 6A $6$ $6$ $C_6\times A_6$ 3B 2A 6A $(7,12,11,8,10,9)$
$S_3^2\times A_6$ 6B $6$ $6$ $C_6\times A_6$ 3A 2B 6B $(7,9,11,8,10,12)$
$S_3^2\times A_6$ 6C $6$ $90$ $C_{12}:D_6$ 3A 2D 6C $(1,6)(4,5)(7,10,11)(8,9,12)$
$S_3^2\times A_6$ 6D $6$ $90$ $C_{12}:D_6$ 3B 2D 6D $(1,4)(2,6)(7,10,11)(8,12,9)$
$S_3^2\times A_6$ 6E $6$ $120$ $C_3^2\times D_6$ 3D 2A 6E $(1,5,2)(7,9)(8,10)(11,12)$
$S_3^2\times A_6$ 6F $6$ $120$ $C_3^2\times D_6$ 3E 2A 6F $(1,6,2)(3,4,5)(7,8)(9,11)(10,12)$
$S_3^2\times A_6$ 6G $6$ $120$ $C_3^2\times D_6$ 3E 2B 6G $(1,2,3)(4,6,5)(7,8)(9,10)(11,12)$
$S_3^2\times A_6$ 6H $6$ $120$ $C_3^2\times D_6$ 3D 2B 6H $(1,4,6)(7,8)(9,10)(11,12)$
$S_3^2\times A_6$ 6I $6$ $180$ $D_4\times C_3^2$ 3C 2D 6I $(2,4)(3,5)(7,11,10)$
$S_3^2\times A_6$ 6J $6$ $240$ $C_3^2\times C_6$ 3F 2B 6J $(1,2,3)(4,6,5)(7,8,10,9,11,12)$
$S_3^2\times A_6$ 6K $6$ $240$ $C_3^2\times C_6$ 3G 2B 6K $(1,2,6)(7,9,11,8,10,12)$
$S_3^2\times A_6$ 6L $6$ $240$ $C_3^2\times C_6$ 3H 2A 6L $(1,6,3)(2,5,4)(7,8,10,12,11,9)$
$S_3^2\times A_6$ 6M $6$ $240$ $C_3^2\times C_6$ 3I 2A 6M $(2,6,4)(7,12,11,8,10,9)$
$S_3^2\times A_6$ 6N $6$ $270$ $C_6\times D_4$ 3A 2E 6N $(1,3)(2,6)(7,8,10,9,11,12)$
$S_3^2\times A_6$ 6O $6$ $270$ $C_6\times D_4$ 3B 2F 6O $(2,5)(4,6)(7,12,10,9,11,8)$
$S_3^2\times A_6$ 6P $6$ $360$ $C_6^2$ 3D 2C 6P $(2,5,3)(7,11)(8,12)$
$S_3^2\times A_6$ 6Q $6$ $360$ $C_6^2$ 3E 2C 6Q $(1,5,2)(3,4,6)(8,9)(10,11)$
$S_3^2\times A_6$ 10A1 $10$ $216$ $S_3\times C_{10}$ 5A2 10A3 2A $(1,2,4,6,3)(7,8)(9,11)(10,12)$
$S_3^2\times A_6$ 10A3 $10$ $216$ $S_3\times C_{10}$ 5A1 10A1 2A $(1,6,2,3,4)(7,8)(9,11)(10,12)$
$S_3^2\times A_6$ 10B1 $10$ $216$ $S_3\times C_{10}$ 5A2 10B3 2B $(1,6,3,2,4)(7,8)(9,10)(11,12)$
$S_3^2\times A_6$ 10B3 $10$ $216$ $S_3\times C_{10}$ 5A1 10B1 2B $(1,2,6,4,3)(7,8)(9,10)(11,12)$
$S_3^2\times A_6$ 10C1 $10$ $648$ $C_2\times C_{10}$ 5A1 10C3 2C $(1,6,2,3,4)(7,10)(8,12)$
$S_3^2\times A_6$ 10C3 $10$ $648$ $C_2\times C_{10}$ 5A2 10C1 2C $(1,3,6,4,2)(7,10)(8,12)$
$S_3^2\times A_6$ 12A $12$ $180$ $S_3\times C_{12}$ 6D 4A 12A $(1,6,4,2)(3,5)(7,11,10)(8,9,12)$
$S_3^2\times A_6$ 12B $12$ $180$ $S_3\times C_{12}$ 6C 4A 12B $(1,2)(3,5,6,4)(7,11,10)(8,12,9)$
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