Today, we've encounter two concepts from group theory(Permutations group and homomorphism). It is instructive to manipulate permutations. Assume that you have To check if your cycle decompositions is correct, plug list thact contains pair (element, its image), in our case
[(1, 2, 3), (4, 5, 6, 7)]
For permutations group, you need to define the group first, then define an element by passing its representaion as string to the created group. Now you can do what ever you like.
(1,2,4,8,5,6,3)
* 1 f f^2 f^3 f^4 f^5 f^6 f^7 f^8 f^9 f^10 f^11 f^12 f^13
+----------------------------------------------------------------------
1| 1 f f^2 f^3 f^4 f^5 f^6 f^7 f^8 f^9 f^10 f^11 f^12 f^13
f| f f^2 f^3 f^4 f^5 f^6 f^7 f^8 f^9 f^10 f^11 f^12 f^13 1
f^2| f^2 f^3 f^4 f^5 f^6 f^7 f^8 f^9 f^10 f^11 f^12 f^13 1 f
f^3| f^3 f^4 f^5 f^6 f^7 f^8 f^9 f^10 f^11 f^12 f^13 1 f f^2
f^4| f^4 f^5 f^6 f^7 f^8 f^9 f^10 f^11 f^12 f^13 1 f f^2 f^3
f^5| f^5 f^6 f^7 f^8 f^9 f^10 f^11 f^12 f^13 1 f f^2 f^3 f^4
f^6| f^6 f^7 f^8 f^9 f^10 f^11 f^12 f^13 1 f f^2 f^3 f^4 f^5
f^7| f^7 f^8 f^9 f^10 f^11 f^12 f^13 1 f f^2 f^3 f^4 f^5 f^6
f^8| f^8 f^9 f^10 f^11 f^12 f^13 1 f f^2 f^3 f^4 f^5 f^6 f^7
f^9| f^9 f^10 f^11 f^12 f^13 1 f f^2 f^3 f^4 f^5 f^6 f^7 f^8
f^10| f^10 f^11 f^12 f^13 1 f f^2 f^3 f^4 f^5 f^6 f^7 f^8 f^9
f^11| f^11 f^12 f^13 1 f f^2 f^3 f^4 f^5 f^6 f^7 f^8 f^9 f^10
f^12| f^12 f^13 1 f f^2 f^3 f^4 f^5 f^6 f^7 f^8 f^9 f^10 f^11
f^13| f^13 1 f f^2 f^3 f^4 f^5 f^6 f^7 f^8 f^9 f^10 f^11 f^12
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* () (1,2) (1,2,3,4) (1,3)(2,4) (1,3,4) (2,3,4) (1,4,3,2) (1,3,4,2) (1,3,2,4) (1,4,2,3) (1,2,4,3) (2,4,3) (1,4,3) (1,4)(2,3) (1,4,2) (1,3,2) (1,3) (3,4) (2,4) (1,4) (2,3) (1,2)(3,4) (1,2,3) (1,2,4)
+------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
()| () (1,2) (1,2,3,4) (1,3)(2,4) (1,3,4) (2,3,4) (1,4,3,2) (1,3,4,2) (1,3,2,4) (1,4,2,3) (1,2,4,3) (2,4,3) (1,4,3) (1,4)(2,3) (1,4,2) (1,3,2) (1,3) (3,4) (2,4) (1,4) (2,3) (1,2)(3,4) (1,2,3) (1,2,4)
(1,2)| (1,2) () (1,3,4) (1,4,2,3) (1,2,3,4) (1,3,4,2) (2,4,3) (2,3,4) (1,4)(2,3) (1,3)(2,4) (1,4,3) (1,4,3,2) (1,2,4,3) (1,3,2,4) (2,4) (2,3) (1,2,3) (1,2)(3,4) (1,4,2) (1,2,4) (1,3,2) (3,4) (1,3) (1,4)
(1,2,3,4)| (1,2,3,4) (2,3,4) (1,3)(2,4) (1,4,3,2) (1,2,4,3) (1,3,2,4) () (2,4,3) (1,4,3) (1,3,2) (1,4,2) (1,4) (1,2) (1,3) (2,3) (3,4) (1,2)(3,4) (1,2,4) (1,4)(2,3) (1,2,3) (1,3,4) (2,4) (1,3,4,2) (1,4,2,3)
(1,3)(2,4)| (1,3)(2,4) (1,3,2,4) (1,4,3,2) () (1,4,2) (1,4,3) (1,2,3,4) (1,4) (1,2) (3,4) (2,3) (1,2,3) (2,3,4) (1,2)(3,4) (1,3,4) (1,2,4) (2,4) (1,4,2,3) (1,3) (1,3,4,2) (1,2,4,3) (1,4)(2,3) (2,4,3) (1,3,2)
(1,3,4)| (1,3,4) (1,3,4,2) (1,4,2,3) (2,4,3) (1,4,3) (1,4)(2,3) (1,2) (1,4,3,2) (1,2,4,3) (2,3) (2,4) (1,2,4) () (1,2,3) (1,3,2) (1,2)(3,4) (3,4) (1,4) (1,3,2,4) (1,3) (1,2,3,4) (1,4,2) (2,3,4) (1,3)(2,4)
(2,3,4)| (2,3,4) (1,2,3,4) (1,2,4,3) (1,3,2) (1,3)(2,4) (2,4,3) (1,4) (1,3,2,4) (1,3) (1,4,3,2) (1,2) () (1,4,2) (1,4,3) (1,4)(2,3) (1,3,4) (1,3,4,2) (2,4) (2,3) (1,4,2,3) (3,4) (1,2,4) (1,2)(3,4) (1,2,3)
(1,4,3,2)| (1,4,3,2) (1,4,3) () (1,2,3,4) (2,3) (1,2) (1,3)(2,4) (1,2,3) (2,3,4) (1,2,4) (1,3,4) (1,3,4,2) (1,3,2,4) (2,4) (1,2,4,3) (1,4,2,3) (1,4)(2,3) (1,3,2) (1,2)(3,4) (2,4,3) (1,4,2) (1,3) (1,4) (3,4)
(1,3,4,2)| (1,3,4,2) (1,3,4) (1,4,3) (2,3) (1,4,2,3) (1,4,3,2) (1,2,4) (1,4)(2,3) (1,2,3) (2,4,3) () (1,2) (2,4) (1,2,4,3) (1,3,2,4) (1,2,3,4) (2,3,4) (1,4,2) (1,3,2) (1,3)(2,4) (1,2)(3,4) (1,4) (3,4) (1,3)
(1,3,2,4)| (1,3,2,4) (1,3)(2,4) (1,4,2) (3,4) (1,4,3,2) (1,4) (1,2,3) (1,4,3) (1,2)(3,4) () (2,3,4) (1,2,3,4) (2,3) (1,2) (1,3) (1,2,4,3) (2,4,3) (1,4)(2,3) (1,3,4) (1,3,2) (1,2,4) (1,4,2,3) (2,4) (1,3,4,2)
(1,4,2,3)| (1,4,2,3) (1,4)(2,3) (2,4,3) (1,2) (2,4) (1,2,4,3) (1,3,4) (1,2,4) () (1,2)(3,4) (1,3,2) (1,3) (1,3,4,2) (3,4) (1,2,3,4) (1,4) (1,4,2) (1,3)(2,4) (1,2,3) (2,3,4) (1,4,3) (1,3,2,4) (1,4,3,2) (2,3)
(1,2,4,3)| (1,2,4,3) (2,4,3) (1,3,2) (1,4) (1,2) (1,3) (2,3,4) () (1,4,2) (1,3,4) (1,4)(2,3) (1,4,2,3) (1,2,3,4) (1,3,4,2) (3,4) (2,4) (1,2,4) (1,2,3) (1,4,3) (1,2)(3,4) (1,3)(2,4) (2,3) (1,3,2,4) (1,4,3,2)
(2,4,3)| (2,4,3) (1,2,4,3) (1,2) (1,3,4) (1,3,2) () (1,4,2,3) (1,3) (1,3,4,2) (1,4) (1,2,3,4) (2,3,4) (1,4)(2,3) (1,4,2) (1,4,3) (1,3)(2,4) (1,3,2,4) (2,3) (3,4) (1,4,3,2) (2,4) (1,2,3) (1,2,4) (1,2)(3,4)
(1,4,3)| (1,4,3) (1,4,3,2) (2,3) (1,2,4) () (1,2,3) (1,3,4,2) (1,2) (2,4) (1,2,3,4) (1,3,2,4) (1,3)(2,4) (1,3,4) (2,3,4) (1,2)(3,4) (1,4,2) (1,4) (1,3) (1,2,4,3) (3,4) (1,4,2,3) (1,3,2) (1,4)(2,3) (2,4,3)
(1,4)(2,3)| (1,4)(2,3) (1,4,2,3) (2,4) (1,2)(3,4) (2,4,3) (1,2,4) (1,3) (1,2,4,3) (3,4) (1,2) (1,3,4,2) (1,3,4) (1,3,2) () (1,2,3) (1,4,3) (1,4,3,2) (1,3,2,4) (1,2,3,4) (2,3) (1,4) (1,3)(2,4) (1,4,2) (2,3,4)
(1,4,2)| (1,4,2) (1,4) (3,4) (1,2,3) (2,3,4) (1,2)(3,4) (1,3,2,4) (1,2,3,4) (2,3) (1,2,4,3) (1,3) (1,3,2) (1,3)(2,4) (2,4,3) (1,2,4) (1,4)(2,3) (1,4,2,3) (1,3,4,2) (1,2) (2,4) (1,4,3,2) (1,3,4) (1,4,3) ()
(1,3,2)| (1,3,2) (1,3) (1,4) (2,3,4) (1,4)(2,3) (1,4,2) (1,2,4,3) (1,4,2,3) (1,2,3,4) (2,4) (3,4) (1,2)(3,4) (2,4,3) (1,2,4) (1,3)(2,4) (1,2,3) (2,3) (1,4,3,2) (1,3,4,2) (1,3,2,4) (1,2) (1,4,3) () (1,3,4)
(1,3)| (1,3) (1,3,2) (1,4)(2,3) (2,4) (1,4) (1,4,2,3) (1,2)(3,4) (1,4,2) (1,2,4) (2,3,4) (2,4,3) (1,2,4,3) (3,4) (1,2,3,4) (1,3,4,2) (1,2) () (1,4,3) (1,3)(2,4) (1,3,4) (1,2,3) (1,4,3,2) (2,3) (1,3,2,4)
(3,4)| (3,4) (1,2)(3,4) (1,2,3) (1,3,2,4) (1,3) (2,3) (1,4,2) (1,3,2) (1,3)(2,4) (1,4)(2,3) (1,2,4) (2,4) (1,4) (1,4,2,3) (1,4,3,2) (1,3,4,2) (1,3,4) () (2,4,3) (1,4,3) (2,3,4) (1,2) (1,2,3,4) (1,2,4,3)
(2,4)| (2,4) (1,2,4) (1,2)(3,4) (1,3) (1,3,4,2) (3,4) (1,4)(2,3) (1,3,4) (1,3,2) (1,4,3) (1,2,3) (2,3) (1,4,2,3) (1,4,3,2) (1,4) (1,3,2,4) (1,3)(2,4) (2,3,4) () (1,4,2) (2,4,3) (1,2,3,4) (1,2,4,3) (1,2)
(1,4)| (1,4) (1,4,2) (2,3,4) (1,2,4,3) (3,4) (1,2,3,4) (1,3,2) (1,2)(3,4) (2,4,3) (1,2,3) (1,3)(2,4) (1,3,2,4) (1,3) (2,3) (1,2) (1,4,3,2) (1,4,3) (1,3,4) (1,2,4) () (1,4)(2,3) (1,3,4,2) (1,4,2,3) (2,4)
(2,3)| (2,3) (1,2,3) (1,2,4) (1,3,4,2) (1,3,2,4) (2,4) (1,4,3) (1,3)(2,4) (1,3,4) (1,4,2) (1,2)(3,4) (3,4) (1,4,3,2) (1,4) (1,4,2,3) (1,3) (1,3,2) (2,4,3) (2,3,4) (1,4)(2,3) () (1,2,4,3) (1,2) (1,2,3,4)
(1,2)(3,4)| (1,2)(3,4) (3,4) (1,3) (1,4)(2,3) (1,2,3) (1,3,2) (2,4) (2,3) (1,4,2,3) (1,3,2,4) (1,4) (1,4,2) (1,2,4) (1,3)(2,4) (2,4,3) (2,3,4) (1,2,3,4) (1,2) (1,4,3,2) (1,2,4,3) (1,3,4,2) () (1,3,4) (1,4,3)
(1,2,3)| (1,2,3) (2,3) (1,3,2,4) (1,4,2) (1,2,4) (1,3)(2,4) (3,4) (2,4) (1,4) (1,3,4,2) (1,4,3,2) (1,4,3) (1,2)(3,4) (1,3,4) (2,3,4) () (1,2) (1,2,4,3) (1,4,2,3) (1,2,3,4) (1,3) (2,4,3) (1,3,2) (1,4)(2,3)
(1,2,4)| (1,2,4) (2,4) (1,3,4,2) (1,4,3) (1,2)(3,4) (1,3,4) (2,3) (3,4) (1,4,3,2) (1,3) (1,4,2,3) (1,4)(2,3) (1,2,3) (1,3,2) () (2,4,3) (1,2,4,3) (1,2,3,4) (1,4) (1,2) (1,3,2,4) (2,3,4) (1,3)(2,4) (1,4,2)
Futurama: Prisoner of Benda
3D rendering not yet implemented