Properties

Label 22T12
Order \(1210\)
n \(22\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Related objects

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Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: None

Low degree siblings

There are no siblings 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 $ $5$ $11$ $( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)(12,17,22,16,21,15,20,14,19,13,18)$
$ 11, 11 $ $5$ $11$ $( 1, 8, 4,11, 7, 3,10, 6, 2, 9, 5)(12,22,21,20,19,18,17,16,15,14,13)$
$ 11, 11 $ $5$ $11$ $( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)(12,21,19,17,15,13,22,20,18,16,14)$
$ 11, 11 $ $5$ $11$ $( 1, 7, 2, 8, 3, 9, 4,10, 5,11, 6)(12,19,15,22,18,14,21,17,13,20,16)$
$ 11, 11 $ $5$ $11$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,15,18,21,13,16,19,22,14,17,20)$
$ 11, 11 $ $5$ $11$ $( 1, 3, 5, 7, 9,11, 2, 4, 6, 8,10)(12,18,13,19,14,20,15,21,16,22,17)$
$ 11, 11 $ $5$ $11$ $( 1, 5, 9, 2, 6,10, 3, 7,11, 4, 8)(12,13,14,15,16,17,18,19,20,21,22)$
$ 11, 11 $ $5$ $11$ $( 1, 9, 6, 3,11, 8, 5, 2,10, 7, 4)(12,14,16,18,20,22,13,15,17,19,21)$
$ 11, 11 $ $5$ $11$ $( 1, 6,11, 5,10, 4, 9, 3, 8, 2, 7)(12,16,20,13,17,21,14,18,22,15,19)$
$ 11, 11 $ $5$ $11$ $( 1,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(12,20,17,14,22,19,16,13,21,18,15)$
$ 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $11$ $(12,17,22,16,21,15,20,14,19,13,18)$
$ 11, 11 $ $10$ $11$ $( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)(12,22,21,20,19,18,17,16,15,14,13)$
$ 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 $ $10$ $11$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,20,17,14,22,19,16,13,21,18,15)$
$ 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $11$ $( 1, 3, 5, 7, 9,11, 2, 4, 6, 8,10)$
$ 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 $ $10$ $11$ $( 1, 5, 9, 2, 6,10, 3, 7,11, 4, 8)(12,22,21,20,19,18,17,16,15,14,13)$
$ 5, 5, 5, 5, 1, 1 $ $242$ $5$ $( 2, 4,10, 6, 5)( 3, 7, 8,11, 9)(13,16,17,21,15)(14,20,22,19,18)$
$ 5, 5, 5, 5, 1, 1 $ $242$ $5$ $( 2,10, 5, 4, 6)( 3, 8, 9, 7,11)(13,17,15,16,21)(14,22,18,20,19)$
$ 22 $ $55$ $22$ $( 1,22, 6,16,11,21, 5,15,10,20, 4,14, 9,19, 3,13, 8,18, 2,12, 7,17)$
$ 22 $ $55$ $22$ $( 1,16, 9,13, 6,21, 3,18,11,15, 8,12, 5,20, 2,17,10,14, 7,22, 4,19)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $55$ $2$ $( 1,21)( 2,22)( 3,12)( 4,13)( 5,14)( 6,15)( 7,16)( 8,17)( 9,18)(10,19)(11,20)$
$ 22 $ $55$ $22$ $( 1,20, 7,15, 2,21, 8,16, 3,22, 9,17, 4,12,10,18, 5,13,11,19, 6,14)$
$ 22 $ $55$ $22$ $( 1,18, 8,14, 4,21,11,17, 7,13, 3,20,10,16, 6,12, 2,19, 9,15, 5,22)$
$ 22 $ $55$ $22$ $( 1,14,10,12, 8,21, 6,19, 4,17, 2,15,11,13, 9,22, 7,20, 5,18, 3,16)$
$ 22 $ $55$ $22$ $( 1,17, 3,19, 5,21, 7,12, 9,14,11,16, 2,18, 4,20, 6,22, 8,13,10,15)$
$ 22 $ $55$ $22$ $( 1,12,11,22,10,21, 9,20, 8,19, 7,18, 6,17, 5,16, 4,15, 3,14, 2,13)$
$ 22 $ $55$ $22$ $( 1,13, 5,17, 9,21, 2,14, 6,18,10,22, 3,15, 7,19,11,12, 4,16, 8,20)$
$ 22 $ $55$ $22$ $( 1,15, 4,18, 7,21,10,13, 2,16, 5,19, 8,22,11,14, 3,17, 6,20, 9,12)$
$ 22 $ $55$ $22$ $( 1,19, 2,20, 3,21, 4,22, 5,12, 6,13, 7,14, 8,15, 9,16,10,17,11,18)$

Group invariants

Order:  $1210=2 \cdot 5 \cdot 11^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [1210, 7]
Character table: Data not available.