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Properties

Label 22T16
Order \(2420\)
n \(22\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
10:  $D_{5}$
20:  20T2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: None

Low degree siblings

44T60

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 $ $1$ $1$ $()$
$ 11, 11 $ $20$ $11$ $( 1, 6,11, 5,10, 4, 9, 3, 8, 2, 7)(12,14,16,18,20,22,13,15,17,19,21)$
$ 11, 11 $ $20$ $11$ $( 1,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(12,16,20,13,17,21,14,18,22,15,19)$
$ 11, 11 $ $20$ $11$ $( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)(12,20,17,14,22,19,16,13,21,18,15)$
$ 11, 11 $ $20$ $11$ $( 1, 8, 4,11, 7, 3,10, 6, 2, 9, 5)(12,17,22,16,21,15,20,14,19,13,18)$
$ 11, 11 $ $20$ $11$ $( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)(12,22,21,20,19,18,17,16,15,14,13)$
$ 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $11$ $(12,16,20,13,17,21,14,18,22,15,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $121$ $2$ $( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,22)(14,21)(15,20)(16,19)(17,18)$
$ 5, 5, 5, 5, 1, 1 $ $242$ $5$ $( 2, 6, 4, 5,10)( 3,11, 7, 9, 8)(13,21,16,15,17)(14,19,20,18,22)$
$ 10, 10, 1, 1 $ $242$ $10$ $( 2, 7, 4, 8,10,11, 6, 9, 5, 3)(13,14,16,20,17,22,21,19,15,18)$
$ 5, 5, 5, 5, 1, 1 $ $242$ $5$ $( 2, 4,10, 6, 5)( 3, 7, 8,11, 9)(13,16,17,21,15)(14,20,22,19,18)$
$ 10, 10, 1, 1 $ $242$ $10$ $( 2, 9,10, 7, 5,11, 4, 3, 6, 8)(13,19,17,14,15,22,16,18,21,20)$
$ 4, 4, 4, 4, 4, 2 $ $605$ $4$ $( 1,16, 3,20)( 2,18)( 4,22,11,14)( 5,13,10,12)( 6,15, 9,21)( 7,17, 8,19)$
$ 4, 4, 4, 4, 4, 2 $ $605$ $4$ $( 1,19, 6,20)( 2,17, 5,22)( 3,15, 4,13)( 7,18,11,21)( 8,16,10,12)( 9,14)$

Group invariants

Order:  $2420=2^{2} \cdot 5 \cdot 11^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table:    
      2  2   .   .   .   .   .   .  2  1   1  1   1  2  2
      5  1   .   .   .   .   .   .  1  1   1  1   1  .  .
     11  2   2   2   2   2   2   2  .  .   .  .   .  .  .

        1a 11a 11b 11c 11d 11e 11f 2a 5a 10a 5b 10b 4a 4b
     2P 1a 11b 11c 11d 11e 11a 11f 1a 5b  5b 5a  5a 2a 2a
     3P 1a 11d 11e 11a 11b 11c 11f 2a 5b 10b 5a 10a 4b 4a
     5P 1a 11e 11a 11b 11c 11d 11f 2a 1a  2a 1a  2a 4a 4b
     7P 1a 11c 11d 11e 11a 11b 11f 2a 5b 10b 5a 10a 4b 4a
    11P 1a  1a  1a  1a  1a  1a  1a 2a 5a 10a 5b 10b 4b 4a

X.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
X.3      1   1   1   1   1   1   1 -1  1  -1  1  -1  G -G
X.4      1   1   1   1   1   1   1 -1  1  -1  1  -1 -G  G
X.5      2   2   2   2   2   2   2 -2  F  -F *F -*F  .  .
X.6      2   2   2   2   2   2   2 -2 *F -*F  F  -F  .  .
X.7      2   2   2   2   2   2   2  2  F   F *F  *F  .  .
X.8      2   2   2   2   2   2   2  2 *F  *F  F   F  .  .
X.9     20  -2  -2  -2  -2  -2   9  .  .   .  .   .  .  .
X.10    20   A   C   E   B   D  -2  .  .   .  .   .  .  .
X.11    20   B   D   A   C   E  -2  .  .   .  .   .  .  .
X.12    20   C   E   B   D   A  -2  .  .   .  .   .  .  .
X.13    20   D   A   C   E   B  -2  .  .   .  .   .  .  .
X.14    20   E   B   D   A   C  -2  .  .   .  .   .  .  .

A = 2*E(11)-2*E(11)^4-E(11)^5-E(11)^6-2*E(11)^7+2*E(11)^10
B = -2*E(11)+2*E(11)^3-E(11)^4-E(11)^7+2*E(11)^8-2*E(11)^10
C = -E(11)+2*E(11)^2-2*E(11)^3-2*E(11)^8+2*E(11)^9-E(11)^10
D = -2*E(11)^2-E(11)^3+2*E(11)^5+2*E(11)^6-E(11)^8-2*E(11)^9
E = -E(11)^2+2*E(11)^4-2*E(11)^5-2*E(11)^6+2*E(11)^7-E(11)^9
F = E(5)^2+E(5)^3
  = (-1-Sqrt(5))/2 = -1-b5
G = -E(4)
  = -Sqrt(-1) = -i