Properties

Label 22T13
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 $t$ :  $13$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,20,12,5,22,2,19,11,6,21)(3,17,13,8,16,4,18,14,7,15)(9,10), (1,21,15,3,8,10)(2,22,16,4,7,9)(5,14,11,6,13,12)(17,20)(18,19)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
660:  $\PSL(2,11)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: $\PSL(2,11)$

Low degree siblings

22T13, 24T2948

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $110$ $3$ $( 3,13,10)( 4,14, 9)( 5,21,11)( 6,22,12)( 7,18,16)( 8,17,15)$
$ 6, 6, 6, 2, 2 $ $110$ $6$ $( 1, 2)( 3,14,10, 4,13, 9)( 5,22,11, 6,21,12)( 7,17,16, 8,18,15)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $55$ $2$ $( 3,10)( 4, 9)( 5,16)( 6,15)( 7,11)( 8,12)(17,22)(18,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $55$ $2$ $( 1, 2)( 3, 9)( 4,10)( 5,15)( 6,16)( 7,12)( 8,11)(13,14)(17,21)(18,22)(19,20)$
$ 10, 10, 2 $ $132$ $10$ $( 1, 2)( 3,17, 5, 8,11, 4,18, 6, 7,12)( 9,16,19,21,14,10,15,20,22,13)$
$ 5, 5, 5, 5, 1, 1 $ $132$ $5$ $( 3,18, 5, 7,11)( 4,17, 6, 8,12)( 9,15,19,22,14)(10,16,20,21,13)$
$ 10, 10, 2 $ $132$ $10$ $( 1, 2)( 3, 9,18,19,21, 4,10,17,20,22)( 5,14,16,12, 7, 6,13,15,11, 8)$
$ 5, 5, 5, 5, 1, 1 $ $132$ $5$ $( 3,10,18,20,21)( 4, 9,17,19,22)( 5,13,16,11, 7)( 6,14,15,12, 8)$
$ 6, 6, 6, 2, 2 $ $110$ $6$ $( 1,20)( 2,19)( 3, 9,13, 4,10,14)( 5, 8,21,17,11,15)( 6, 7,22,18,12,16)$
$ 6, 6, 3, 3, 2, 2 $ $110$ $6$ $( 1,19)( 2,20)( 3,10,13)( 4, 9,14)( 5, 7,21,18,11,16)( 6, 8,22,17,12,15)$
$ 22 $ $60$ $22$ $( 1,20,22,18, 6, 3,15,11, 9,13, 8, 2,19,21,17, 5, 4,16,12,10,14, 7)$
$ 11, 11 $ $60$ $11$ $( 1,19,22,17, 6, 4,15,12, 9,14, 8)( 2,20,21,18, 5, 3,16,11,10,13, 7)$
$ 22 $ $60$ $22$ $( 1,20,22, 3,14,16, 6,10, 8,11,17, 2,19,21, 4,13,15, 5, 9, 7,12,18)$
$ 11, 11 $ $60$ $11$ $( 1,19,22, 4,14,15, 6, 9, 8,12,17)( 2,20,21, 3,13,16, 5,10, 7,11,18)$

Group invariants

Order:  $1320=2^{3} \cdot 3 \cdot 5 \cdot 11$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  [1320, 134]
Character table:   
      2  3   3  2  2  3  3   1   1   1   1  2  2   1   1   1   1
      3  1   1  1  1  1  1   .   .   .   .  1  1   .   .   .   .
      5  1   1  .  .  .  .   1   1   1   1  .  .   .   .   .   .
     11  1   1  .  .  .  .   .   .   .   .  .  .   1   1   1   1

        1a  2a 3a 6a 2b 2c 10a  5a 10b  5b 6b 6c 22a 11a 22b 11b
     2P 1a  1a 3a 3a 1a 1a  5b  5b  5a  5a 3a 3a 11b 11b 11a 11a
     3P 1a  2a 1a 2a 2b 2c 10b  5b 10a  5a 2c 2b 22a 11a 22b 11b
     5P 1a  2a 3a 6a 2b 2c  2a  1a  2a  1a 6b 6c 22a 11a 22b 11b
     7P 1a  2a 3a 6a 2b 2c 10b  5b 10a  5a 6b 6c 22b 11b 22a 11a
    11P 1a  2a 3a 6a 2b 2c 10a  5a 10b  5b 6b 6c  2a  1a  2a  1a
    13P 1a  2a 3a 6a 2b 2c 10b  5b 10a  5a 6b 6c 22b 11b 22a 11a
    17P 1a  2a 3a 6a 2b 2c 10b  5b 10a  5a 6b 6c 22b 11b 22a 11a
    19P 1a  2a 3a 6a 2b 2c 10a  5a 10b  5b 6b 6c 22b 11b 22a 11a

X.1      1   1  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   1  -1   1
X.3      5   5 -1 -1  1  1   .   .   .   .  1  1   B   B  /B  /B
X.4      5   5 -1 -1  1  1   .   .   .   .  1  1  /B  /B   B   B
X.5      5  -5 -1  1  1 -1   .   .   .   . -1  1  -B   B -/B  /B
X.6      5  -5 -1  1  1 -1   .   .   .   . -1  1 -/B  /B  -B   B
X.7     10  10  1  1 -2 -2   .   .   .   .  1  1  -1  -1  -1  -1
X.8     10 -10  1 -1 -2  2   .   .   .   . -1  1   1  -1   1  -1
X.9     10  10  1  1  2  2   .   .   .   . -1 -1  -1  -1  -1  -1
X.10    10 -10  1 -1  2 -2   .   .   .   .  1 -1   1  -1   1  -1
X.11    11  11 -1 -1 -1 -1   1   1   1   1 -1 -1   .   .   .   .
X.12    11 -11 -1  1 -1  1  -1   1  -1   1  1 -1   .   .   .   .
X.13    12 -12  .  .  .  .   A  -A  *A -*A  .  .  -1   1  -1   1
X.14    12 -12  .  .  .  .  *A -*A   A  -A  .  .  -1   1  -1   1
X.15    12  12  .  .  .  . -*A -*A  -A  -A  .  .   1   1   1   1
X.16    12  12  .  .  .  .  -A  -A -*A -*A  .  .   1   1   1   1

A = -E(5)-E(5)^4
  = (1-Sqrt(5))/2 = -b5
B = E(11)+E(11)^3+E(11)^4+E(11)^5+E(11)^9
  = (-1+Sqrt(-11))/2 = b11