Properties

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

Related objects

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Group action invariants

Degree $n$ :  $11$
Transitive number $t$ :  $4$
Group :  $F_{11}$
CHM label :  $F_{110}(11)=11:10$
Parity:  $-1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,3,4,5,6,7,8,9,10,11), (1,2,4,8,5,10,9,7,3,6)
$|\Aut(F/K)|$:  $1$

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

Prime degree - none

Low degree siblings

22T4

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

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 10a  5a  5b  5c 10b 10c 10d  5d 2a 11a
     2P 1a  5b  5d  5c  5a  5a  5c  5d  5b 1a 11a
     3P 1a 10d  5c  5d  5b 10c 10a 10b  5a 2a 11a
     5P 1a  2a  1a  1a  1a  2a  2a  2a  1a 2a 11a
     7P 1a 10c  5d  5c  5a 10d 10b 10a  5b 2a 11a
    11P 1a 10a  5a  5b  5c 10b 10c 10d  5d 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 -/B  -B -/A  /A   B  /B  -A -1   1
X.4      1   B  -A -/A -/B  /B  /A   A  -B -1   1
X.5      1  /B -/A  -A  -B   B   A  /A -/B -1   1
X.6      1  /A  -B -/B  -A   A  /B   B -/A -1   1
X.7      1 -/A  -B -/B  -A  -A -/B  -B -/A  1   1
X.8      1 -/B -/A  -A  -B  -B  -A -/A -/B  1   1
X.9      1  -B  -A -/A -/B -/B -/A  -A  -B  1   1
X.10     1  -A -/B  -B -/A -/A  -B -/B  -A  1   1
X.11    10   .   .   .   .   .   .   .   .  .  -1

A = -E(5)
B = -E(5)^2