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Properties

Label 22T4
Order \(110\)
n \(22\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $F_{11}$

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Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: $F_{11}$

Low degree siblings

11T4

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

Group invariants

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

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

X.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
X.3      1  A /A /B  B  -B -/B -/A  -A -1   1
X.4      1  B /B  A /A -/A  -A -/B  -B -1   1
X.5      1 /B  B /A  A  -A -/A  -B -/B -1   1
X.6      1 /A  A  B /B -/B  -B  -A -/A -1   1
X.7      1  A /A /B  B   B  /B  /A   A  1   1
X.8      1  B /B  A /A  /A   A  /B   B  1   1
X.9      1 /B  B /A  A   A  /A   B  /B  1   1
X.10     1 /A  A  B /B  /B   B   A  /A  1   1
X.11    10  .  .  .  .   .   .   .   .  .  -1

A = E(5)^4
B = E(5)^3