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

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  5  5  4  4  4  4  4  4  1   1   1   1   1   1  2  2  2  2
      3  2   2  1  1  .  .  .  .  .  .  .   .   .   .   .   .  2  2  1  1
      5  1   1  .  .  .  .  .  .  .  .  1   1   .   .   .   .  .  .  .  .
     11  1   1  .  .  .  .  .  .  .  .  .   .   1   1   1   1  .  .  .  .

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

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  2  2  2  2  .  .  .  .  .   .  -1  -1  -1  -1  1  1 -1 -1
X.4     10 -10  2 -2 -2  2  .  .  .  .  .   .   1  -1   1  -1  1 -1  1 -1
X.5     10 -10 -2  2  .  .  A -A -A  A  .   .   1  -1   1  -1  1 -1 -1  1
X.6     10 -10 -2  2  .  . -A  A  A -A  .   .   1  -1   1  -1  1 -1 -1  1
X.7     10  10 -2 -2  .  .  A  A -A -A  .   .  -1  -1  -1  -1  1  1  1  1
X.8     10  10 -2 -2  .  . -A -A  A  A  .   .  -1  -1  -1  -1  1  1  1  1
X.9     11  11  3  3 -1 -1 -1 -1 -1 -1  1   1   .   .   .   .  2  2  .  .
X.10    11 -11  3 -3  1 -1 -1  1 -1  1  1  -1   .   .   .   .  2 -2  .  .
X.11    16 -16  .  .  .  .  .  .  .  .  1  -1   B  -B  /B -/B -2  2  .  .
X.12    16 -16  .  .  .  .  .  .  .  .  1  -1  /B -/B   B  -B -2  2  .  .
X.13    16  16  .  .  .  .  .  .  .  .  1   1 -/B -/B  -B  -B -2 -2  .  .
X.14    16  16  .  .  .  .  .  .  .  .  1   1  -B  -B -/B -/B -2 -2  .  .
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 -3 -3  1  1 -1 -1 -1 -1  .   .   1   1   1   1  .  .  .  .
X.18    45 -45 -3  3 -1  1 -1  1 -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(8)-E(8)^3
  = -Sqrt(-2) = -i2
B = -E(11)-E(11)^3-E(11)^4-E(11)^5-E(11)^9
  = (1-Sqrt(-11))/2 = -b11