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Properties

Label 12T272
Order \(7920\)
n \(12\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $M_{11}$

Related objects

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Group action invariants

Degree $n$ :  $12$
Transitive number $t$ :  $272$
Group :  $M_{11}$
CHM label :  $M_{11}(12)$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,7,3,10,5,9,6,12)(2,11,8,4), (1,6,3,9)(2,7,12,10,4,5,11,8)
$|\Aut(F/K)|$:  $1$

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

11T6, 22T22

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 $ $1$ $1$ $()$
$ 5, 5, 1, 1 $ $1584$ $5$ $( 1, 3, 6, 7,10)( 2, 9, 8,11, 5)$
$ 2, 2, 2, 2, 1, 1, 1, 1 $ $165$ $2$ $( 1, 4)( 2,11)( 5, 8)( 6, 7)$
$ 3, 3, 3, 1, 1, 1 $ $440$ $3$ $( 1, 2, 6)( 4,11, 7)( 9,10,12)$
$ 6, 3, 2, 1 $ $1320$ $6$ $( 1, 7, 2, 4, 6,11)( 5, 8)( 9,12,10)$
$ 11, 1 $ $720$ $11$ $( 1, 7,12, 4, 3,10, 5, 8,11, 6, 2)$
$ 11, 1 $ $720$ $11$ $( 1, 2, 6,11, 8, 5,10, 3, 4,12, 7)$
$ 4, 4, 2, 2 $ $990$ $4$ $( 1, 9)( 2,11)( 3, 7, 8,12)( 4, 6,10, 5)$
$ 8, 4 $ $990$ $8$ $( 1, 2, 9,11)( 3,10, 7, 5, 8, 4,12, 6)$
$ 8, 4 $ $990$ $8$ $( 1,11, 9, 2)( 3, 6,12, 4, 8, 5, 7,10)$

Group invariants

Order:  $7920=2^{4} \cdot 3^{2} \cdot 5 \cdot 11$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table:    
      2  4   .   .  4  3  3  3  .  1  1
      3  2   .   .  1  .  .  .  .  2  1
      5  1   .   .  .  .  .  .  1  .  .
     11  1   1   1  .  .  .  .  .  .  .

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

X.1      1   1   1  1  1  1  1  1  1  1
X.2     10  -1  -1  2  2  .  .  .  1 -1
X.3     10  -1  -1 -2  .  B -B  .  1  1
X.4     10  -1  -1 -2  . -B  B  .  1  1
X.5     11   .   .  3 -1 -1 -1  1  2  .
X.6     16   A  /A  .  .  .  .  1 -2  .
X.7     16  /A   A  .  .  .  .  1 -2  .
X.8     44   .   .  4  .  .  . -1 -1  1
X.9     45   1   1 -3  1 -1 -1  .  .  .
X.10    55   .   . -1 -1  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
B = -E(8)-E(8)^3
  = -Sqrt(-2) = -i2