Properties

Label 22T17
Order \(2420\)
n \(22\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Learn more about

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
10:  $D_{5}$
20:  $D_{10}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: None

Low degree siblings

44T61

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, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 11, 11 $ $10$ $11$ $( 1,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(12,13,14,15,16,17,18,19,20,21,22)$
$ 11, 11 $ $10$ $11$ $( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)(12,14,16,18,20,22,13,15,17,19,21)$
$ 11, 11 $ $10$ $11$ $( 1, 8, 4,11, 7, 3,10, 6, 2, 9, 5)(12,16,20,13,17,21,14,18,22,15,19)$
$ 11, 11 $ $10$ $11$ $( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)(12,20,17,14,22,19,16,13,21,18,15)$
$ 11, 11 $ $10$ $11$ $( 1, 7, 2, 8, 3, 9, 4,10, 5,11, 6)(12,17,22,16,21,15,20,14,19,13,18)$
$ 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $20$ $11$ $(12,13,14,15,16,17,18,19,20,21,22)$
$ 11, 11 $ $10$ $11$ $( 1,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(12,14,16,18,20,22,13,15,17,19,21)$
$ 11, 11 $ $10$ $11$ $( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)(12,15,18,21,13,16,19,22,14,17,20)$
$ 11, 11 $ $10$ $11$ $( 1, 7, 2, 8, 3, 9, 4,10, 5,11, 6)(12,18,13,19,14,20,15,21,16,22,17)$
$ 11, 11 $ $10$ $11$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,13,14,15,16,17,18,19,20,21,22)$
$ 11, 11 $ $10$ $11$ $( 1, 3, 5, 7, 9,11, 2, 4, 6, 8,10)(12,14,16,18,20,22,13,15,17,19,21)$
$ 5, 5, 5, 5, 1, 1 $ $242$ $5$ $( 2, 5, 6,10, 4)( 3, 9,11, 8, 7)(13,15,21,17,16)(14,18,19,22,20)$
$ 5, 5, 5, 5, 1, 1 $ $242$ $5$ $( 2, 6, 4, 5,10)( 3,11, 7, 9, 8)(13,21,16,15,17)(14,19,20,18,22)$
$ 10, 10, 1, 1 $ $242$ $10$ $( 2, 3, 5, 9, 6,11,10, 8, 4, 7)(13,18,15,19,21,22,17,20,16,14)$
$ 10, 10, 1, 1 $ $242$ $10$ $( 2, 9,10, 7, 5,11, 4, 3, 6, 8)(13,19,17,14,15,22,16,18,21,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $121$ $2$ $( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,22)(14,21)(15,20)(16,19)(17,18)$
$ 22 $ $110$ $22$ $( 1,16, 6,20,11,13, 5,17,10,21, 4,14, 9,18, 3,22, 8,15, 2,19, 7,12)$
$ 22 $ $110$ $22$ $( 1,17, 9,19, 6,21, 3,12,11,14, 8,16, 5,18, 2,20,10,22, 7,13, 4,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $55$ $2$ $( 1,18)( 2,21)( 3,13)( 4,16)( 5,19)( 6,22)( 7,14)( 8,17)( 9,20)(10,12)(11,15)$
$ 22 $ $110$ $22$ $( 1,13, 8,12, 4,22,11,21, 7,20, 3,19,10,18, 6,17, 2,16, 9,15, 5,14)$
$ 22 $ $110$ $22$ $( 1,21,10,15, 8,20, 6,14, 4,19, 2,13,11,18, 9,12, 7,17, 5,22, 3,16)$
$ 22 $ $110$ $22$ $( 1,14,11,22,10,19, 9,16, 8,13, 7,21, 6,18, 5,15, 4,12, 3,20, 2,17)$
$ 22 $ $110$ $22$ $( 1,14, 6,16,11,18, 5,20,10,22, 4,13, 9,15, 3,17, 8,19, 2,21, 7,12)$
$ 22 $ $110$ $22$ $( 1,15, 2,22, 3,18, 4,14, 5,21, 6,17, 7,13, 8,20, 9,16,10,12,11,19)$
$ 22 $ $110$ $22$ $( 1,16, 9,17, 6,18, 3,19,11,20, 8,21, 5,22, 2,12,10,13, 7,14, 4,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $55$ $2$ $( 1,18)( 2,14)( 3,21)( 4,17)( 5,13)( 6,20)( 7,16)( 8,12)( 9,19)(10,15)(11,22)$
$ 22 $ $110$ $22$ $( 1,19, 8,13, 4,18,11,12, 7,17, 3,22,10,16, 6,21, 2,15, 9,20, 5,14)$
$ 22 $ $110$ $22$ $( 1,13,10,21, 8,18, 6,15, 4,12, 2,20,11,17, 9,14, 7,22, 5,19, 3,16)$

Group invariants

Order:  $2420=2^{2} \cdot 5 \cdot 11^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.