Properties

Label 22T21
Order \(6050\)
n \(22\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
5:  $C_5$
10:  $D_{5}$, $C_{10}$
50:  $D_5\times C_5$

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 $ $50$ $11$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,22,21,20,19,18,17,16,15,14,13)$
$ 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $11$ $(12,16,20,13,17,21,14,18,22,15,19)$
$ 11, 11 $ $25$ $11$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,15,18,21,13,16,19,22,14,17,20)$
$ 11, 11 $ $25$ $11$ $( 1, 3, 5, 7, 9,11, 2, 4, 6, 8,10)(12,14,16,18,20,22,13,15,17,19,21)$
$ 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $10$ $11$ $(12,20,17,14,22,19,16,13,21,18,15)$
$ 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 $ $121$ $5$ $( 2, 5, 6,10, 4)( 3, 9,11, 8, 7)(13,16,17,21,15)(14,20,22,19,18)$
$ 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $22$ $5$ $(13,15,21,17,16)(14,18,19,22,20)$
$ 11, 5, 5, 1 $ $110$ $55$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,22,19,21,16)(13,14,17,15,20)$
$ 11, 5, 5, 1 $ $110$ $55$ $( 1, 3, 5, 7, 9,11, 2, 4, 6, 8,10)(12,21,15,19,20)(14,16,22,18,17)$
$ 5, 5, 5, 5, 1, 1 $ $242$ $5$ $( 2, 4,10, 6, 5)( 3, 7, 8,11, 9)(13,21,16,15,17)(14,19,20,18,22)$
$ 5, 5, 5, 5, 1, 1 $ $242$ $5$ $( 2,10, 5, 4, 6)( 3, 8, 9, 7,11)(13,17,15,16,21)(14,22,18,20,19)$
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $22$ $5$ $( 2, 5, 6,10, 4)( 3, 9,11, 8, 7)$
$ 11, 5, 5, 1 $ $110$ $55$ $( 1, 2, 6,11, 9)( 3,10, 5, 7, 4)(12,22,21,20,19,18,17,16,15,14,13)$
$ 11, 5, 5, 1 $ $110$ $55$ $( 1, 3,11,10, 6)( 2, 7, 5, 8, 9)(12,21,19,17,15,13,22,20,18,16,14)$
$ 5, 5, 5, 5, 1, 1 $ $242$ $5$ $( 2, 6, 4, 5,10)( 3,11, 7, 9, 8)(13,16,17,21,15)(14,20,22,19,18)$
$ 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $22$ $5$ $(13,21,16,15,17)(14,19,20,18,22)$
$ 11, 5, 5, 1 $ $110$ $55$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,22,13,20,17)(14,18,21,15,16)$
$ 11, 5, 5, 1 $ $110$ $55$ $( 1, 3, 5, 7, 9,11, 2, 4, 6, 8,10)(12,21,14,17,22)(13,19,18,20,16)$
$ 5, 5, 5, 5, 1, 1 $ $242$ $5$ $( 2, 4,10, 6, 5)( 3, 7, 8,11, 9)(13,17,15,16,21)(14,22,18,20,19)$
$ 5, 5, 5, 5, 1, 1 $ $242$ $5$ $( 2,10, 5, 4, 6)( 3, 8, 9, 7,11)(13,16,17,21,15)(14,20,22,19,18)$
$ 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $22$ $5$ $( 2, 6, 4, 5,10)( 3,11, 7, 9, 8)$
$ 11, 5, 5, 1 $ $110$ $55$ $( 1, 2, 7,10, 3)( 4, 6, 5,11, 8)(12,22,21,20,19,18,17,16,15,14,13)$
$ 11, 5, 5, 1 $ $110$ $55$ $( 1, 3, 2, 8, 5)( 4, 7,11, 9,10)(12,21,19,17,15,13,22,20,18,16,14)$
$ 10, 10, 2 $ $605$ $10$ $( 1,16,10,17, 4,20, 8,18, 9,12)( 2,21)( 3,15, 5,14,11,22, 7,13, 6,19)$
$ 10, 10, 2 $ $605$ $10$ $( 1,13, 5,18, 3,21, 4,14, 9,12)( 2,17,10,16, 6,22, 8,19, 7,15)(11,20)$
$ 22 $ $275$ $22$ $( 1,15, 4,18, 7,21,10,13, 2,16, 5,19, 8,22,11,14, 3,17, 6,20, 9,12)$
$ 22 $ $275$ $22$ $( 1,19, 8,15, 4,22,11,18, 7,14, 3,21,10,17, 6,13, 2,20, 9,16, 5,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $55$ $2$ $( 1,12)( 2,13)( 3,14)( 4,15)( 5,16)( 6,17)( 7,18)( 8,19)( 9,20)(10,21)(11,22)$
$ 10, 10, 2 $ $605$ $10$ $( 1,17, 2,15, 6,18,11,19, 9,12)( 3,13,10,21, 5,20, 7,16, 4,22)( 8,14)$
$ 10, 10, 2 $ $605$ $10$ $( 1,21, 6,14, 7,17, 5,22, 9,12)( 2,13, 4,19,11,18, 8,20, 3,16)(10,15)$

Group invariants

Order:  $6050=2 \cdot 5^{2} \cdot 11^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.