# Properties

 Label 11T4 Degree $11$ Order $110$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $F_{11}$

# Related objects

## 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) $|\Aut(F/K)|$: $1$ Generators: (1,2,3,4,5,6,7,8,9,10,11), (1,2,4,8,5,10,9,7,3,6)

## 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 Type Size Order Representative $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $10, 1$ $11$ $10$ $( 2, 3, 5, 9, 6,11,10, 8, 4, 7)$ $5, 5, 1$ $11$ $5$ $( 2, 4,10, 6, 5)( 3, 7, 8,11, 9)$ $5, 5, 1$ $11$ $5$ $( 2, 5, 6,10, 4)( 3, 9,11, 8, 7)$ $5, 5, 1$ $11$ $5$ $( 2, 6, 4, 5,10)( 3,11, 7, 9, 8)$ $10, 1$ $11$ $10$ $( 2, 7, 4, 8,10,11, 6, 9, 5, 3)$ $10, 1$ $11$ $10$ $( 2, 8, 6, 3, 4,11, 5, 7,10, 9)$ $10, 1$ $11$ $10$ $( 2, 9,10, 7, 5,11, 4, 3, 6, 8)$ $5, 5, 1$ $11$ $5$ $( 2,10, 5, 4, 6)( 3, 8, 9, 7,11)$ $2, 2, 2, 2, 2, 1$ $11$ $2$ $( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)$ $11$ $10$ $11$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)$

## 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