Properties

Label 22T5
Order \(110\)
n \(22\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times C_{11}:C_5$

Related objects

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Group action invariants

Degree $n$ :  $22$
Transitive number $t$ :  $5$
Group :  $C_2\times C_{11}:C_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,16,7,22,13,5,19,12,3,18,10,2,15,8,21,14,6,20,11,4,17,9), (1,11,17,3,21)(2,12,18,4,22)(5,9,8,20,14)(6,10,7,19,13)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
5:  $C_5$
10:  $C_{10}$
55:  $C_{11}:C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: $C_{11}:C_5$

Low degree siblings

There are no siblings 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 $ $1$ $1$ $()$
$ 5, 5, 5, 5, 1, 1 $ $11$ $5$ $( 3, 7,19,11,10)( 4, 8,20,12, 9)( 5,14,16,22,18)( 6,13,15,21,17)$
$ 5, 5, 5, 5, 1, 1 $ $11$ $5$ $( 3,10,11,19, 7)( 4, 9,12,20, 8)( 5,18,22,16,14)( 6,17,21,15,13)$
$ 5, 5, 5, 5, 1, 1 $ $11$ $5$ $( 3,11, 7,10,19)( 4,12, 8, 9,20)( 5,22,14,18,16)( 6,21,13,17,15)$
$ 5, 5, 5, 5, 1, 1 $ $11$ $5$ $( 3,19,10, 7,11)( 4,20, 9, 8,12)( 5,16,18,14,22)( 6,15,17,13,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)$
$ 10, 10, 2 $ $11$ $10$ $( 1, 2)( 3, 8,19,12,10, 4, 7,20,11, 9)( 5,13,16,21,18, 6,14,15,22,17)$
$ 10, 10, 2 $ $11$ $10$ $( 1, 2)( 3, 9,11,20, 7, 4,10,12,19, 8)( 5,17,22,15,14, 6,18,21,16,13)$
$ 10, 10, 2 $ $11$ $10$ $( 1, 2)( 3,12, 7, 9,19, 4,11, 8,10,20)( 5,21,14,17,16, 6,22,13,18,15)$
$ 10, 10, 2 $ $11$ $10$ $( 1, 2)( 3,20,10, 8,11, 4,19, 9, 7,12)( 5,15,18,13,22, 6,16,17,14,21)$
$ 11, 11 $ $5$ $11$ $( 1, 3, 6, 7,10,11,13,15,17,19,21)( 2, 4, 5, 8, 9,12,14,16,18,20,22)$
$ 22 $ $5$ $22$ $( 1, 4, 6, 8,10,12,13,16,17,20,21, 2, 3, 5, 7, 9,11,14,15,18,19,22)$
$ 22 $ $5$ $22$ $( 1, 5,10,14,17,22, 3, 8,11,16,19, 2, 6, 9,13,18,21, 4, 7,12,15,20)$
$ 11, 11 $ $5$ $11$ $( 1, 6,10,13,17,21, 3, 7,11,15,19)( 2, 5, 9,14,18,22, 4, 8,12,16,20)$

Group invariants

Order:  $110=2 \cdot 5 \cdot 11$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [110, 2]
Character table:   
      2  1  1  1  1  1  1   1   1   1   1   1   1   1   1
      5  1  1  1  1  1  1   1   1   1   1   .   .   .   .
     11  1  .  .  .  .  1   .   .   .   .   1   1   1   1

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

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

A = E(5)^4
B = E(5)^3
C = E(11)^2+E(11)^6+E(11)^7+E(11)^8+E(11)^10
  = (-1-Sqrt(-11))/2 = -1-b11