Properties

Label 22T15
Order \(2420\)
n \(22\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

Learn more about

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
5:  $C_5$
10:  $C_{10}$ x 3
20:  20T3
110:  $F_{11}$ x 2
220:  22T6 x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: None

Low degree siblings

22T15 x 4, 44T59 x 5

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

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.