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Properties

Label 11T2
Order \(22\)
n \(11\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $D_{11}$

Related objects

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Group action invariants

Degree $n$ :  $11$
Transitive number $t$ :  $2$
Group :  $D_{11}$
CHM label :  $D(11)=11:2$
Parity:  $-1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,3,4,5,6,7,8,9,10,11), (1,10)(2,9)(3,8)(4,7)(5,6)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

22T2

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 1 $ $11$ $2$ $( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)$
$ 11 $ $2$ $11$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)$
$ 11 $ $2$ $11$ $( 1, 3, 5, 7, 9,11, 2, 4, 6, 8,10)$
$ 11 $ $2$ $11$ $( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)$
$ 11 $ $2$ $11$ $( 1, 5, 9, 2, 6,10, 3, 7,11, 4, 8)$
$ 11 $ $2$ $11$ $( 1, 6,11, 5,10, 4, 9, 3, 8, 2, 7)$

Group invariants

Order:  $22=2 \cdot 11$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [22, 1]
Character table:    
     2  1  1   .   .   .   .   .
    11  1  .   1   1   1   1   1

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

X.1     1  1   1   1   1   1   1
X.2     1 -1   1   1   1   1   1
X.3     2  .   A   D   E   B   C
X.4     2  .   B   E   A   C   D
X.5     2  .   C   A   B   D   E
X.6     2  .   D   B   C   E   A
X.7     2  .   E   C   D   A   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