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

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  1  4   1  5  3  2  2
      3   3   .   .  1  3  1  .  .  .  1   .  .  .  2  1
      5   1   .   .  .  .  .  .  .  1  1   1  .  .  .  .
     11   1   1   1  .  .  .  .  .  .  .   .  .  .  .  .

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

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