Properties

Label 22T14
Order \(1320\)
n \(22\)
Cyclic No
Abelian No
Solvable No
Primitive No
$p$-group No

Related objects

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: None

Low degree siblings

12T218, 24T2949

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 $ $55$ $2$ $( 3, 4)( 5, 7)( 6, 9)( 8,11)(14,15)(16,18)(17,20)(19,22)$
$ 4, 4, 4, 4, 2, 2, 2 $ $110$ $4$ $( 1,13)( 2,12)( 3,16, 4,18)( 5,14, 7,15)( 6,20, 9,17)( 8,19,11,22)(10,21)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $110$ $3$ $( 1, 2,10)( 3, 7, 9)( 4, 5, 6)(12,21,13)(14,20,18)(15,17,16)$
$ 6, 6, 3, 3, 2, 2 $ $110$ $6$ $( 1,10, 2)( 3, 6, 7, 4, 9, 5)( 8,11)(12,13,21)(14,16,20,15,18,17)(19,22)$
$ 12, 6, 4 $ $110$ $12$ $( 1,12,10,13, 2,21)( 3,14, 6,16, 7,20, 4,15, 9,18, 5,17)( 8,22,11,19)$
$ 12, 6, 4 $ $110$ $12$ $( 1,12,10,13, 2,21)( 3,15, 6,18, 7,17, 4,14, 9,16, 5,20)( 8,19,11,22)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $66$ $2$ $( 1,21)( 2,12)( 3,20)( 4,17)( 5,15)( 6,16)( 7,14)( 8,19)( 9,18)(10,13)(11,22)$
$ 5, 5, 5, 5, 1, 1 $ $132$ $5$ $( 1, 7, 9, 6, 5)( 2,11, 3, 4, 8)(12,22,20,17,19)(14,18,16,15,21)$
$ 5, 5, 5, 5, 1, 1 $ $132$ $5$ $( 1, 6, 7, 5, 9)( 2, 4,11, 8, 3)(12,17,22,19,20)(14,15,18,21,16)$
$ 10, 10, 2 $ $132$ $10$ $( 1,14, 9,16, 5,21, 7,18, 6,15)( 2,22, 3,17, 8,12,11,20, 4,19)(10,13)$
$ 10, 10, 2 $ $132$ $10$ $( 1,18, 5,14, 6,21, 9,15, 7,16)( 2,20, 8,22, 4,12, 3,19,11,17)(10,13)$
$ 11, 11 $ $120$ $11$ $( 1, 7, 4, 5,11,10, 6, 2, 3, 8, 9)(12,20,19,14,13,17,21,22,16,15,18)$

Group invariants

Order:  $1320=2^{3} \cdot 3 \cdot 5 \cdot 11$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [1320, 133]
Character table:   
      2  3  2  1  1   1   1  3  2  2  2   .   2   2
      3  1  .  .  .   .   .  1  1  1  1   .   1   1
      5  1  1  1  1   1   1  .  .  .  .   .   .   .
     11  1  .  .  .   .   .  .  .  .  .   1   .   .

        1a 2a 5a 5b 10a 10b 2b 3a 6a 4a 11a 12a 12b
     2P 1a 1a 5b 5a  5a  5b 1a 3a 3a 2b 11a  6a  6a
     3P 1a 2a 5b 5a 10b 10a 2b 1a 2b 4a 11a  4a  4a
     5P 1a 2a 1a 1a  2a  2a 2b 3a 6a 4a 11a 12b 12a
     7P 1a 2a 5b 5a 10b 10a 2b 3a 6a 4a 11a 12b 12a
    11P 1a 2a 5a 5b 10a 10b 2b 3a 6a 4a  1a 12a 12b

X.1      1  1  1  1   1   1  1  1  1  1   1   1   1
X.2      1 -1  1  1  -1  -1  1  1  1 -1   1  -1  -1
X.3     10  .  .  .   .   . -2  1  1 -2  -1   1   1
X.4     10  .  .  .   .   . -2  1  1  2  -1  -1  -1
X.5     10  .  .  .   .   .  2 -2  2  .  -1   .   .
X.6     10  .  .  .   .   .  2  1 -1  .  -1   B  -B
X.7     10  .  .  .   .   .  2  1 -1  .  -1  -B   B
X.8     11 -1  1  1  -1  -1 -1 -1 -1  1   .   1   1
X.9     11  1  1  1   1   1 -1 -1 -1 -1   .  -1  -1
X.10    12 -2  A *A -*A  -A  .  .  .  .   1   .   .
X.11    12 -2 *A  A  -A -*A  .  .  .  .   1   .   .
X.12    12  2  A *A  *A   A  .  .  .  .   1   .   .
X.13    12  2 *A  A   A  *A  .  .  .  .   1   .   .

A = E(5)^2+E(5)^3
  = (-1-Sqrt(5))/2 = -1-b5
B = -E(12)^7+E(12)^11
  = Sqrt(3) = r3