# Properties

 Label 22T11 Degree $22$ Order $1210$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no

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## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$10$:  $C_{10}$
$55$:  $C_{11}:C_5$
$110$:  $F_{11}$, 22T5

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 11: None

## Low degree siblings

22T11 x 4

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $11, 11$ $10$ $11$ $( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)(12,16,20,13,17,21,14,18,22,15,19)$ $11, 11$ $10$ $11$ $( 1, 8, 4,11, 7, 3,10, 6, 2, 9, 5)(12,20,17,14,22,19,16,13,21,18,15)$ $11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $10$ $11$ $(12,13,14,15,16,17,18,19,20,21,22)$ $11, 11$ $10$ $11$ $( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)(12,17,22,16,21,15,20,14,19,13,18)$ $11, 11$ $5$ $11$ $( 1, 8, 4,11, 7, 3,10, 6, 2, 9, 5)(12,21,19,17,15,13,22,20,18,16,14)$ $11, 11$ $10$ $11$ $( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)(12,18,13,19,14,20,15,21,16,22,17)$ $11, 11$ $10$ $11$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,22,21,20,19,18,17,16,15,14,13)$ $11, 11$ $10$ $11$ $( 1, 5, 9, 2, 6,10, 3, 7,11, 4, 8)(12,16,20,13,17,21,14,18,22,15,19)$ $11, 11$ $10$ $11$ $( 1, 9, 6, 3,11, 8, 5, 2,10, 7, 4)(12,19,15,22,18,14,21,17,13,20,16)$ $11, 11$ $10$ $11$ $( 1,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(12,15,18,21,13,16,19,22,14,17,20)$ $11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $10$ $11$ $(12,14,16,18,20,22,13,15,17,19,21)$ $11, 11$ $5$ $11$ $( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)(12,19,15,22,18,14,21,17,13,20,16)$ $11, 11$ $10$ $11$ $( 1, 7, 2, 8, 3, 9, 4,10, 5,11, 6)(12,13,14,15,16,17,18,19,20,21,22)$ $5, 5, 5, 5, 1, 1$ $121$ $5$ $( 2, 6, 4, 5,10)( 3,11, 7, 9, 8)(13,17,15,16,21)(14,22,18,20,19)$ $5, 5, 5, 5, 1, 1$ $121$ $5$ $( 2, 4,10, 6, 5)( 3, 7, 8,11, 9)(13,15,21,17,16)(14,18,19,22,20)$ $5, 5, 5, 5, 1, 1$ $121$ $5$ $( 2,10, 5, 4, 6)( 3, 8, 9, 7,11)(13,21,16,15,17)(14,19,20,18,22)$ $5, 5, 5, 5, 1, 1$ $121$ $5$ $( 2, 5, 6,10, 4)( 3, 9,11, 8, 7)(13,16,17,21,15)(14,20,22,19,18)$ $10, 10, 2$ $121$ $10$ $( 1,18,10,14,11,16, 5,15, 8,21)( 2,20, 4,13, 3,22, 9,12, 6,17)( 7,19)$ $10, 10, 2$ $121$ $10$ $( 1,20, 5,16,10,22, 8,13,11,21)( 2,19, 9,12, 4,17, 6,15, 3,18)( 7,14)$ $22$ $55$ $22$ $( 1,19, 8,17, 4,15,11,13, 7,22, 3,20,10,18, 6,16, 2,14, 9,12, 5,21)$ $22$ $55$ $22$ $( 1,13,10,12, 8,22, 6,21, 4,20, 2,19,11,18, 9,17, 7,16, 5,15, 3,14)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $11$ $2$ $( 1,18)( 2,13)( 3,19)( 4,14)( 5,20)( 6,15)( 7,21)( 8,16)( 9,22)(10,17)(11,12)$ $10, 10, 2$ $121$ $10$ $( 1,22, 7,20, 6,13, 8,16, 4,21)( 2,18, 5,17,10,19,11,15, 9,12)( 3,14)$ $10, 10, 2$ $121$ $10$ $( 1,14, 8,15, 7,18, 4,16, 6,21)( 2,22,11,17, 5,13, 9,12,10,20)( 3,19)$

## Group invariants

 Order: $1210=2 \cdot 5 \cdot 11^{2}$ Cyclic: no Abelian: no Solvable: yes GAP id: [1210, 12]
 Character table: not available.