Properties

Label 11T3
Order \(55\)
n \(11\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $C_{11}:C_5$

Related objects

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Group action invariants

Degree $n$ :  $11$
Transitive number $t$ :  $3$
Group :  $C_{11}:C_5$
CHM label :  $F_{55}(11)=11:5$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3,9,5,4)(2,6,7,10,8), (1,2,3,4,5,6,7,8,9,10,11)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
5:  $C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

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

Group invariants

Order:  $55=5 \cdot 11$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [55, 1]
Character table:   
     5  1  1  1  1  1   .   .
    11  1  .  .  .  .   1   1

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

X.1     1  1  1  1  1   1   1
X.2     1  A /A /B  B   1   1
X.3     1  B /B  A /A   1   1
X.4     1 /B  B /A  A   1   1
X.5     1 /A  A  B /B   1   1
X.6     5  .  .  .  .   C  /C
X.7     5  .  .  .  .  /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