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

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

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

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

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