Properties

Label 22T6
Order \(220\)
n \(22\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2\times F_{11}$

Related objects

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Group action invariants

Degree $n$ :  $22$
Transitive number $t$ :  $6$
Group :  $C_2\times F_{11}$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,19,17,6,21,7,11,13,3,10)(2,20,18,5,22,8,12,14,4,9), (1,2)(3,12,7,9,19,4,11,8,10,20)(5,21,14,17,16,6,22,13,18,15)
$|\Aut(F/K)|$:  $2$

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:  $F_{11}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: $F_{11}$

Low degree siblings

22T6, 44T14

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

Group invariants

Order:  $220=2^{2} \cdot 5 \cdot 11$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [220, 7]
Character table: Data not available.