Properties

Label 22T27
Order \(15840\)
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 $t$ :  $27$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,22,17,16,12,5,3,7,14,9,19,2,21,18,15,11,6,4,8,13,10,20), (1,4,10,5,19,22)(2,3,9,6,20,21)(7,17,13,8,18,14)(11,15)(12,16)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
7920:  $M_{11}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: $M_{11}$

Low degree siblings

22T26, 24T12204, 44T140

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

Group invariants

Order:  $15840=2^{5} \cdot 3^{2} \cdot 5 \cdot 11$
Cyclic:  No
Abelian:  No
Solvable:  No
GAP id:  Data not available
Character table:   
      2  5   5   1   1   1   1  5  5  2  2  2  2  4  4  4  4  4  4   1  1
      3  2   2   .   .   .   .  1  1  2  2  1  1  .  .  .  .  .  .   .  .
      5  1   1   .   .   .   .  .  .  .  .  .  .  .  .  .  .  .  .   1  1
     11  1   1   1   1   1   1  .  .  .  .  .  .  .  .  .  .  .  .   .  .

        1a  2a 22a 11a 11b 22b 2b 2c 6a 3a 6b 6c 4a 4b 8a 8b 8c 8d 10a 5a
     2P 1a  1a 11b 11b 11a 11a 1a 1a 3a 3a 3a 3a 2b 2b 4a 4a 4a 4a  5a 5a
     3P 1a  2a 22a 11a 11b 22b 2b 2c 2a 1a 2b 2c 4a 4b 8a 8b 8c 8d 10a 5a
     5P 1a  2a 22a 11a 11b 22b 2b 2c 6a 3a 6b 6c 4a 4b 8c 8d 8a 8b  2a 1a
     7P 1a  2a 22b 11b 11a 22a 2b 2c 6a 3a 6b 6c 4a 4b 8c 8d 8a 8b 10a 5a
    11P 1a  2a  2a  1a  1a  2a 2b 2c 6a 3a 6b 6c 4a 4b 8a 8b 8c 8d 10a 5a
    13P 1a  2a 22b 11b 11a 22a 2b 2c 6a 3a 6b 6c 4a 4b 8c 8d 8a 8b 10a 5a
    17P 1a  2a 22b 11b 11a 22a 2b 2c 6a 3a 6b 6c 4a 4b 8a 8b 8c 8d 10a 5a
    19P 1a  2a 22b 11b 11a 22a 2b 2c 6a 3a 6b 6c 4a 4b 8a 8b 8c 8d 10a 5a

X.1      1   1   1   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 -1  1  -1  1
X.3     10  10  -1  -1  -1  -1  2  2  1  1 -1 -1  2  2  .  .  .  .   .  .
X.4     10 -10   1  -1  -1   1  2 -2 -1  1 -1  1  2 -2  .  .  .  .   .  .
X.5     10 -10   1  -1  -1   1 -2  2 -1  1  1 -1  .  .  B -B -B  B   .  .
X.6     10 -10   1  -1  -1   1 -2  2 -1  1  1 -1  .  . -B  B  B -B   .  .
X.7     10  10  -1  -1  -1  -1 -2 -2  1  1  1  1  .  .  B  B -B -B   .  .
X.8     10  10  -1  -1  -1  -1 -2 -2  1  1  1  1  .  . -B -B  B  B   .  .
X.9     11  11   .   .   .   .  3  3  2  2  .  . -1 -1 -1 -1 -1 -1   1  1
X.10    11 -11   .   .   .   .  3 -3 -2  2  .  . -1  1  1 -1  1 -1  -1  1
X.11    16 -16   A  -A -/A  /A  .  .  2 -2  .  .  .  .  .  .  .  .  -1  1
X.12    16 -16  /A -/A  -A   A  .  .  2 -2  .  .  .  .  .  .  .  .  -1  1
X.13    16  16 -/A -/A  -A  -A  .  . -2 -2  .  .  .  .  .  .  .  .   1  1
X.14    16  16  -A  -A -/A -/A  .  . -2 -2  .  .  .  .  .  .  .  .   1  1
X.15    44  44   .   .   .   .  4  4 -1 -1  1  1  .  .  .  .  .  .  -1 -1
X.16    44 -44   .   .   .   .  4 -4  1 -1  1 -1  .  .  .  .  .  .   1 -1
X.17    45  45   1   1   1   1 -3 -3  .  .  .  .  1  1 -1 -1 -1 -1   .  .
X.18    45 -45  -1   1   1  -1 -3  3  .  .  .  .  1 -1  1 -1  1 -1   .  .
X.19    55  55   .   .   .   . -1 -1  1  1 -1 -1 -1 -1  1  1  1  1   .  .
X.20    55 -55   .   .   .   . -1  1 -1  1 -1  1 -1  1 -1  1 -1  1   .  .

A = -E(11)-E(11)^3-E(11)^4-E(11)^5-E(11)^9
  = (1-Sqrt(-11))/2 = -b11
B = -E(8)-E(8)^3
  = -Sqrt(-2) = -i2