# Properties

 Label 22T4 Degree $22$ Order $110$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $F_{11}$

# Related objects

## Group action invariants

 Degree $n$: $22$ Transitive number $t$: $4$ Group: $F_{11}$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $2$ 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)

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

## 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