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Properties

Label 22T3
Order \(44\)
n \(22\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{22}$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
22:  $D_{11}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: $D_{11}$

Low degree siblings

22T3, 44T4

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

Group invariants

Order:  $44=2^{2} \cdot 11$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [44, 3]
Character table:    
      2  2  2  2  2   1   1   1   1   1   1   1   1   1   1
     11  1  .  1  .   1   1   1   1   1   1   1   1   1   1

        1a 2a 2b 2c 22a 11a 22b 11b 11c 22c 22d 11d 22e 11e
     2P 1a 1a 1a 1a 11b 11b 11d 11d 11e 11e 11c 11c 11a 11a
     3P 1a 2a 2b 2c 22c 11c 22e 11e 11b 22b 22a 11a 22d 11d
     5P 1a 2a 2b 2c 22e 11e 22a 11a 11d 22d 22b 11b 22c 11c
     7P 1a 2a 2b 2c 22d 11d 22c 11c 11a 22a 22e 11e 22b 11b
    11P 1a 2a 2b 2c  2b  1a  2b  1a  1a  2b  2b  1a  2b  1a
    13P 1a 2a 2b 2c 22b 11b 22d 11d 11e 22e 22c 11c 22a 11a
    17P 1a 2a 2b 2c 22e 11e 22a 11a 11d 22d 22b 11b 22c 11c
    19P 1a 2a 2b 2c 22c 11c 22e 11e 11b 22b 22a 11a 22d 11d

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

A = E(11)^4+E(11)^7
B = E(11)^5+E(11)^6
C = E(11)^2+E(11)^9
D = E(11)^3+E(11)^8
E = E(11)+E(11)^10