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Properties

Label 11T6
Order \(7920\)
n \(11\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $M_{11}$

Related objects

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Group action invariants

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

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

12T272, 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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1, 1 $ $165$ $2$ $( 1, 6)( 3, 8)( 4, 7)( 5,11)$
$ 4, 4, 1, 1, 1 $ $990$ $4$ $( 1, 5, 6,11)( 3, 4, 8, 7)$
$ 8, 2, 1 $ $990$ $8$ $( 1, 4, 5, 8, 6, 7,11, 3)( 9,10)$
$ 8, 2, 1 $ $990$ $8$ $( 1, 3,11, 7, 6, 8, 5, 4)( 9,10)$
$ 3, 3, 3, 1, 1 $ $440$ $3$ $( 1, 7, 4)( 2,10, 8)( 5,11, 6)$
$ 6, 3, 2 $ $1320$ $6$ $( 1, 5, 7,11, 4, 6)( 2, 8,10)( 3, 9)$
$ 11 $ $720$ $11$ $( 1, 7, 2, 8, 3, 9, 4,10, 5,11, 6)$
$ 11 $ $720$ $11$ $( 1, 6,11, 5,10, 4, 9, 3, 8, 2, 7)$
$ 5, 5, 1 $ $1584$ $5$ $( 2,10, 5, 4, 6)( 3, 8, 9, 7,11)$

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

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

X.1      1  1  1  1  1  1  1  1   1   1
X.2     10  .  2  2  1 -1  .  .  -1  -1
X.3     10  . -2  .  1  1  A -A  -1  -1
X.4     10  . -2  .  1  1 -A  A  -1  -1
X.5     11  1  3 -1  2  . -1 -1   .   .
X.6     16  1  .  . -2  .  .  .   B  /B
X.7     16  1  .  . -2  .  .  .  /B   B
X.8     44 -1  4  . -1  1  .  .   .   .
X.9     45  . -3  1  .  . -1 -1   1   1
X.10    55  . -1 -1  1 -1  1  1   .   .

A = -E(8)-E(8)^3
  = -Sqrt(-2) = -i2
B = E(11)^2+E(11)^6+E(11)^7+E(11)^8+E(11)^10
  = (-1-Sqrt(-11))/2 = -1-b11