# Properties

 Label 22T2 Degree $22$ Order $22$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $D_{11}$

# Related objects

## Group action invariants

 Degree $n$: $22$ Transitive number $t$: $2$ Group: $D_{11}$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $22$ Generators: (1,19)(2,20)(3,18)(4,17)(5,16)(6,15)(7,14)(8,13)(9,12)(10,11)(21,22), (1,4,6,8,10,12,14,16,18,20,21)(2,3,5,7,9,11,13,15,17,19,22)

## Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 11: $D_{11}$

## Low degree siblings

11T2

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

## 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