Properties

Label 22T22
Degree $22$
Order $7920$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $M_{11}$

Related objects

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Group action invariants

Degree $n$:  $22$
Transitive number $t$:  $22$
Group:  $M_{11}$
Parity:  $1$
Primitive:  no
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $2$
Generators:  (1,11,3,19,14)(2,12,4,20,13)(5,17,16,9,7)(6,18,15,10,8), (1,22,15,4,20)(2,21,16,3,19)(5,12,14,9,8)(6,11,13,10,7)

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 11: $M_{11}$

Low degree siblings

11T6, 12T272

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, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $165$ $2$ $( 1,17)( 2,18)( 5, 8)( 6, 7)( 9,21)(10,22)(11,20)(12,19)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $440$ $3$ $( 1, 5,21)( 2, 6,22)( 3,15,13)( 4,16,14)( 7,10,18)( 8, 9,17)$
$ 6, 6, 3, 3, 2, 2 $ $1320$ $6$ $( 1, 9, 5,17,21, 8)( 2,10, 6,18,22, 7)( 3,13,15)( 4,14,16)(11,20)(12,19)$
$ 11, 11 $ $720$ $11$ $( 1, 5,13,11,17, 9,21, 8,15, 3,20)( 2, 6,14,12,18,10,22, 7,16, 4,19)$
$ 11, 11 $ $720$ $11$ $( 1,20, 3,15, 8,21, 9,17,11,13, 5)( 2,19, 4,16, 7,22,10,18,12,14, 6)$
$ 5, 5, 5, 5, 1, 1 $ $1584$ $5$ $( 1, 4,21,14, 7)( 2, 3,22,13, 8)( 5,20,16,10,18)( 6,19,15, 9,17)$
$ 4, 4, 4, 4, 2, 2, 1, 1 $ $990$ $4$ $( 3,20,11,18)( 4,19,12,17)( 5,14,22, 9)( 6,13,21,10)( 7, 8)(15,16)$
$ 8, 8, 4, 2 $ $990$ $8$ $( 1, 2)( 3, 5,20,14,11,22,18, 9)( 4, 6,19,13,12,21,17,10)( 7,16, 8,15)$
$ 8, 8, 4, 2 $ $990$ $8$ $( 1, 2)( 3, 9,18,22,11,14,20, 5)( 4,10,17,21,12,13,19, 6)( 7,15, 8,16)$

Group invariants

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

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

X.1      1  1  1  1  1  1  1  1   1   1
X.2     10  2  1 -1  .  2  .  .  -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  3  2  .  1 -1 -1 -1   .   .
X.6     16  . -2  .  1  .  .  .   B  /B
X.7     16  . -2  .  1  .  .  .  /B   B
X.8     44  4 -1  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