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I Permutation group and character table

  1. Mar 29, 2017 #1
    Hi everybody,

    I work currently with permutation group, and with the good advice of this forum I discover GAP software (https://www.gap-system.org/) which is an excellent tools for working with group.
    My question is about something that is too strange for me: I have a permutation group G composed of 36 elements (a non abelian group). When I calculate with GAP the character table of G, it has 18 classes of congugacy, each one with order of:
    $$1, 4, 1, 4, 9, 9, 2, 2, 2, 2, 3, 6, 3, 6, 3, 6, 3, 6$$
    It implicates now that G has ##2\times36=72## elements !!!
    Is it normal or have I missed something?

    Thank you much.

    Konte.
     
  2. jcsd
  3. Mar 29, 2017 #2

    Orodruin

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    Each element in the group belongs to exactly one conjugacy class so the sum of the orders of the conjugacy classes should be the order of the group. Without more specifics, it is difficult to analyse your situation.
     
  4. Mar 29, 2017 #3
    Thank you for your answer.

    - I have a group G of exactly 36 elements (permutations).
    - When I ask the GAP program to show me the orders of the conjugacy classes, it gives me the following answer: $$1, 4, 1, 4, 9, 9, 2, 2, 2, 2, 3, 6, 3, 6, 3, 6, 3, 6$$
    - So as you can see, the sum of those numbers exceed 36.

    If necessary, I can give on new post the set of permutation that compose the group.

    Konte
     
    Last edited: Mar 29, 2017
  5. Mar 29, 2017 #4

    Orodruin

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    As I said, without more information about what group you are considering and what conjugacy classes are quoted by your program, we cannot get any further.
     
  6. Mar 29, 2017 #5
    Ok, the 36 elements are permutations:
    a0= id.
    a1=(1,17)(2,18)(3,16)(4,15)(5,14)(6,13)(7,12)(8,11)(9,10)
    a2=(7,3)(8,4)(11,15)(12,16)(19,20)
    a3=(1,17)(2,18)(3,12)(4,11)(5,14)(6,13)(7,16)(8,15)(9,10)(19,20)
    a4=(3,5,7)(4,6,8)
    a5=(1,17)(2,18)(3,14,5,12,7,16)(4,13,6,11,8,15)(9,10)
    a6=(3,5)(4,6)(11,15)(12,16)(19,20)
    a7=(1,17)(2,18)(3,14,5,16,7,12)(4,13,6,15,8,11)(9,10)(19,20)
    a8=(3,7,5)(4,8,6)
    a9=(1,17)(2,18)(3,12,7,14,5,16)(4,11,8,13,6,15)(9,10)
    a10=(5,7)(6,8)(11,15)(12,16)(19,20)
    a11=(1,17)(2,18)(3,16,7,14,5,12)(4,15,8,13,6,11)(9,10)(19,20)
    a12=(11,13,15)(12,14,16)
    a13=(1,17)(2,18)(3,16,7,12,5,14)(4,15,8,11,6,13)
    a14=(3,7)(4,8)(11,13)(12,14)(19,20)
    a15=(1,17)(2,18)(3,12,5,14,7,16)(4,11,6,13,8,15)(9,10)(19,20)
    a16=(3,5,7)(4,6,8)(11,13,15)(12,14,16)
    a17=(1,17)(2,18)(3,14)(4,13)(5,12)(6,11)(7,16)(8,15)(9,10)
    a18=(3,5)(4,6)(11,13)(12,14)(19,20)
    a19=(1,17)(2,18)(3,14,7,12,5,16)(4,13,8,11,6,15)(9,10)(19,20)
    a20=(3,7,5)(4,8,6)(11,13,15)(12,14,16)
    a21=(1,17)(2,18)(3,12,5,16,7,14)(4,11,6,15,8,13)(9,10)
    a22=(5,7)(6,8)(11,13)(12,14)(19,20)
    a23=(1,17)(2,18)(3,16)(4,15)(5,12)(6,11)(7,14)(8,13)(9,10)(19,20)
    a24=(11,15,13)(12,16,14)
    a25=(1,17)(2,18)(3,16,5,14,7,12)(4,15,6,13,8,11)(9,10)
    a26=(3,7)(4,8)(13,15)(14,16)(19,20)
    a27=(1,17)(2,18)(3,12,7,16,5,14)(4,11,8,15,6,13)(9,10)(19,20)
    a28=(3,5,7)(4,6,8)(11,15,13)(12,16,14)
    a29=(1,17)(2,18)(3,14,7,16,5,12)(4,13,8,15,6,11)(9,10)
    a30=(3,5)(4,6)(13,15)(14,16)(19,20)
    a31=(1,17)(2,18)(3,14)(4,13)(5,16)(6,15)(7,12)(8,11)(9,10)(19,20)
    a32=(3,7,5)(4,8,6)(11,15,13)(12,16,14)
    a33=(1,17)(2,18)(3,12)(4,11)(5,16)(6,15)(7,14)(8,13)(9,10)
    a34=(5,7)(6,8)(13,15)(14,16)(19,20)
    a35=(1,17)(2,18)(3,16,5,12,7,14)(4,15,6,11,8,13)(9,10)(19,20)

    The character table given by the program is:

    char.png

    The size of each congugacy classes:## 1, 4, 1, 4, 9, 9, 2, 2, 2, 2, 3, 6, 3, 6, 3, 6, 3, 6##

    The elements of the 18 conjugacy classes given by the program are:
    id.,
    (11,13,15)(12,14,16),
    (9,10),
    (9,10)(11,13,15)(12,14,16),
    (5,7)(6,8)(13,15)(14,16)(19,20),
    (5,7)(6,8)(9,10)(13,15)(14,16)(19,20),
    (3,5,7)(4,6,8)(11,13,15)(12,14,16),
    (3,5,7)(4,6,8)(11,15,13)(12,16,14),
    (3,5,7)(4,6,8)(9,10)(11,13,15)(12,14,16),
    (3,5,7)(4,6,8)(9,10)(11,15,13)(12,16,14),
    (1,17)(2,18)(3,12)(4,11)(5,14)(6,13)(7,16)(8,15)(19,20),
    (1,17)(2,18)(3,12,5,14,7,16)(4,11,6,13,8,15)(19,20),
    (1,17)(2,18)(3,12)(4,11)(5,14)(6,13)(7,16)(8,15)(9,10)(19,20),
    (1,17)(2,18)(3,12,5,14,7,16)(4,11,6,13,8,15)(9,10)(19,20),
    (1,17)(2,18)(3,12)(4,11)(5,16)(6,15)(7,14)(8,13),
    (1,17)(2,18)(3,12,5,16,7,14)(4,11,6,15,8,13),
    (1,17)(2,18)(3,12)(4,11)(5,16)(6,15)(7,14)(8,13)(9,10),
    (1,17)(2,18)(3,12,5,16,7,14)(4,11,6,15,8,13)(9,10),


    Thanks.
    Konte
     
  7. Mar 29, 2017 #6

    Orodruin

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    Assuming that your quoted conjugacy classes are just given a single element of each conjugacy class:

    This conjugacy class is here represented by an element that is not in your listing of the group elements. Of course, I have not checked the closure of your group operation, but it would seem to me that the program thinks you have elements in your group that you do not think that you have.
     
  8. Mar 29, 2017 #7
    Thank you for your answer, it really helps me. Now, I will re-check the group if all elements are there.
    Konte.
     
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