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Properties

Label 12T295
Order \(95040\)
n \(12\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $M_{12}$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $295$
Group :  $M_{12}$
CHM label :  $M(12)$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,5,12,11,8,2,4)(6,10), (1,11,2,3,4)(5,8,12,6,7)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: None

Degree 4: None

Degree 6: None

Low degree siblings

12T295

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $495$ $2$ $( 1, 8)( 2, 7)( 6,10)(11,12)$
$ 4, 4, 1, 1, 1, 1 $ $2970$ $4$ $( 1, 7, 8, 2)( 6,12,10,11)$
$ 8, 2, 1, 1 $ $11880$ $8$ $( 1,10, 7,11, 8, 6, 2,12)( 4, 5)$
$ 11, 1 $ $8640$ $11$ $( 1,12, 6, 2,10, 7, 4,11, 9, 5, 3)$
$ 11, 1 $ $8640$ $11$ $( 1, 3, 5, 9,11, 4, 7,10, 2, 6,12)$
$ 4, 4, 2, 2 $ $2970$ $4$ $( 1, 7, 8, 2)( 3, 9)( 4, 5)( 6,11,10,12)$
$ 8, 4 $ $11880$ $8$ $( 1,10, 7,12, 8, 6, 2,11)( 3, 5, 9, 4)$
$ 5, 5, 1, 1 $ $9504$ $5$ $( 1, 5,11, 4, 8)( 2,12, 7, 3, 9)$
$ 2, 2, 2, 2, 2, 2 $ $396$ $2$ $( 1, 9)( 2, 5)( 3, 8)( 4, 7)( 6,10)(11,12)$
$ 10, 2 $ $9504$ $10$ $( 1,12, 8, 2, 4, 9,11, 3, 5, 7)( 6,10)$
$ 3, 3, 3, 3 $ $2640$ $3$ $( 1, 8,11)( 2, 7, 6)( 3,12, 9)( 4,10, 5)$
$ 6, 6 $ $7920$ $6$ $( 1, 6, 8, 2,11, 7)( 3, 5,12, 4, 9,10)$
$ 3, 3, 3, 1, 1, 1 $ $1760$ $3$ $( 1, 9, 5)( 3, 4, 8)(10,11,12)$
$ 6, 3, 2, 1 $ $15840$ $6$ $( 1,12, 9,10, 5,11)( 2, 7)( 3, 8, 4)$

Group invariants

Order:  $95040=2^{6} \cdot 3^{3} \cdot 5 \cdot 11$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table:    
      2   6  6  1  1  5  3  4  2  2  5   .   .  1   1  3
      3   3  1  3  1  .  .  1  2  1  .   .   .  .   .  .
      5   1  .  .  .  .  .  1  .  .  .   .   .  1   1  .
     11   1  .  .  .  .  .  .  .  .  .   1   1  .   .  .

         1a 2a 3a 6a 4a 8a 2b 3b 6b 4b 11a 11b 5a 10a 8b
     2P  1a 1a 3a 3a 2a 4a 1a 3b 3b 2a 11b 11a 5a  5a 4b
     3P  1a 2a 1a 2a 4a 8a 2b 1a 2b 4b 11a 11b 5a 10a 8b
     5P  1a 2a 3a 6a 4a 8a 2b 3b 6b 4b 11a 11b 1a  2b 8b
     7P  1a 2a 3a 6a 4a 8a 2b 3b 6b 4b 11b 11a 5a 10a 8b
    11P  1a 2a 3a 6a 4a 8a 2b 3b 6b 4b  1a  1a 5a 10a 8b

X.1       1  1  1  1  1  1  1  1  1  1   1   1  1   1  1
X.2      11  3  2  . -1 -1 -1 -1 -1  3   .   .  1  -1  1
X.3      11  3  2  .  3  1 -1 -1 -1 -1   .   .  1  -1 -1
X.4      16  . -2  .  .  .  4  1  1  .   A  /A  1  -1  .
X.5      16  . -2  .  .  .  4  1  1  .  /A   A  1  -1  .
X.6      45 -3  .  .  1 -1  5  3 -1  1   1   1  .   . -1
X.7      54  6  .  .  2  .  6  .  .  2  -1  -1 -1   1  .
X.8      55  7  1  1 -1 -1 -5  1  1 -1   .   .  .   . -1
X.9      55 -1  1 -1 -1  1 -5  1  1  3   .   .  .   . -1
X.10     55 -1  1 -1  3 -1 -5  1  1 -1   .   .  .   .  1
X.11     66  2  3 -1 -2  .  6  .  . -2   .   .  1   1  .
X.12     99  3  .  . -1  1 -1  3 -1 -1   .   . -1  -1  1
X.13    120 -8  3  1  .  .  .  .  .  .  -1  -1  .   .  .
X.14    144  .  .  .  .  .  4 -3  1  .   1   1 -1  -1  .
X.15    176  . -4  .  .  . -4 -1 -1  .   .   .  1   1  .

A = E(11)^2+E(11)^6+E(11)^7+E(11)^8+E(11)^10
  = (-1-Sqrt(-11))/2 = -1-b11