Properties

Label 22T34
Order \(112640\)
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 $t$ :  $34$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (3,16,12,6,7,22,10,13,20,17)(4,15,11,5,8,21,9,14,19,18), (1,15,14,7,12,2,16,13,8,11)(3,21,9,18,20)(4,22,10,17,19)(5,6)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
5:  $C_5$
10:  $C_{10}$
110:  $F_{11}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 11: $F_{11}$

Low degree siblings

22T35, 44T312, 44T313, 44T316

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $11$ $2$ $( 1, 2)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $55$ $2$ $( 3, 4)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $110$ $2$ $( 1, 2)( 5, 6)( 7, 8)( 9,10)(13,14)(17,18)(19,20)(21,22)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $55$ $2$ $( 1, 2)( 3, 4)(11,12)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $110$ $2$ $( 7, 8)( 9,10)(13,14)(17,18)(19,20)(21,22)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $110$ $2$ $( 1, 2)( 3, 4)( 5, 6)(11,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $55$ $2$ $( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $110$ $2$ $( 1, 2)( 3, 4)( 5, 6)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $110$ $2$ $( 9,10)(13,14)(15,16)(17,18)(19,20)(21,22)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $110$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(11,12)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $55$ $2$ $( 9,10)(11,12)(13,14)(17,18)(19,20)(21,22)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $55$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $55$ $2$ $( 5, 6)( 7, 8)( 9,10)(17,18)(19,20)(21,22)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $22$ $2$ $( 1, 2)( 5, 6)( 7, 8)(11,12)(13,14)(15,16)$
$ 11, 11 $ $10240$ $11$ $( 1,19,15,11, 8, 3,21,18,13,10, 5)( 2,20,16,12, 7, 4,22,17,14, 9, 6)$
$ 5, 5, 5, 5, 1, 1 $ $2816$ $5$ $( 3,12, 7,10,20)( 4,11, 8, 9,19)( 5,21,14,18,15)( 6,22,13,17,16)$
$ 10, 5, 5, 2 $ $5632$ $10$ $( 1, 2)( 3,11, 7, 9,20)( 4,12, 8,10,19)( 5,22,14,17,15, 6,21,13,18,16)$
$ 10, 10, 1, 1 $ $2816$ $10$ $( 3,12, 7,10,20, 4,11, 8, 9,19)( 5,21,14,18,16, 6,22,13,17,15)$
$ 5, 5, 5, 5, 1, 1 $ $2816$ $5$ $( 3, 7,20,12,10)( 4, 8,19,11, 9)( 5,14,15,21,18)( 6,13,16,22,17)$
$ 10, 5, 5, 2 $ $5632$ $10$ $( 1, 2)( 3, 8,20,11,10)( 4, 7,19,12, 9)( 5,13,15,22,18, 6,14,16,21,17)$
$ 10, 10, 1, 1 $ $2816$ $10$ $( 3, 7,20,12,10, 4, 8,19,11, 9)( 5,14,16,22,17, 6,13,15,21,18)$
$ 5, 5, 5, 5, 1, 1 $ $2816$ $5$ $( 3,20,10, 7,12)( 4,19, 9, 8,11)( 5,15,18,14,21)( 6,16,17,13,22)$
$ 10, 5, 5, 2 $ $5632$ $10$ $( 1, 2)( 3,19,10, 8,12)( 4,20, 9, 7,11)( 5,16,18,13,21, 6,15,17,14,22)$
$ 10, 10, 1, 1 $ $2816$ $10$ $( 3,20,10, 7,12, 4,19, 9, 8,11)( 5,16,17,13,22, 6,15,18,14,21)$
$ 5, 5, 5, 5, 1, 1 $ $2816$ $5$ $( 3,10,12,20, 7)( 4, 9,11,19, 8)( 5,18,21,15,14)( 6,17,22,16,13)$
$ 10, 5, 5, 2 $ $5632$ $10$ $( 1, 2)( 3, 9,12,19, 7)( 4,10,11,20, 8)( 5,17,21,16,14, 6,18,22,15,13)$
$ 10, 10, 1, 1 $ $2816$ $10$ $( 3,10,12,20, 7, 4, 9,11,19, 8)( 5,18,21,16,13, 6,17,22,15,14)$
$ 10, 10, 1, 1 $ $5632$ $10$ $( 3,16,12, 6, 7,22,10,13,20,17)( 4,15,11, 5, 8,21, 9,14,19,18)$
$ 20, 2 $ $5632$ $20$ $( 1, 2)( 3,15,12, 5, 7,21,10,14,20,18, 4,16,11, 6, 8,22, 9,13,19,17)$
$ 10, 10, 1, 1 $ $5632$ $10$ $( 3, 6,10,17,12,22,20,16, 7,13)( 4, 5, 9,18,11,21,19,15, 8,14)$
$ 20, 2 $ $5632$ $20$ $( 1, 2)( 3, 5,10,18,12,21,20,15, 7,14, 4, 6, 9,17,11,22,19,16, 8,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $352$ $2$ $( 3,22)( 4,21)( 5,19)( 6,20)( 7,17)( 8,18)( 9,15)(10,16)(11,14)(12,13)$
$ 4, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1760$ $4$ $( 1, 2)( 3,21, 4,22)( 5,20)( 6,19)( 7,18)( 8,17)( 9,16)(10,15)(11,13)(12,14)$
$ 4, 4, 2, 2, 2, 2, 2, 2, 1, 1 $ $1760$ $4$ $( 3,22, 4,21)( 5,19)( 6,20)( 7,17)( 8,18)( 9,16,10,15)(11,14)(12,13)$
$ 4, 4, 4, 2, 2, 2, 2, 2 $ $1760$ $4$ $( 1, 2)( 3,21, 4,22)( 5,20)( 6,19)( 7,18)( 8,17)( 9,15,10,16)(11,13,12,14)$
$ 4, 4, 2, 2, 2, 2, 2, 2, 1, 1 $ $1760$ $4$ $( 3,22, 4,21)( 5,19)( 6,20)( 7,17)( 8,18)( 9,15)(10,16)(11,14,12,13)$
$ 4, 4, 4, 4, 2, 2, 1, 1 $ $1760$ $4$ $( 3,21, 4,22)( 5,20, 6,19)( 7,18)( 8,17)( 9,15,10,16)(11,13,12,14)$
$ 4, 4, 4, 2, 2, 2, 2, 2 $ $1760$ $4$ $( 1, 2)( 3,22, 4,21)( 5,19, 6,20)( 7,17)( 8,18)( 9,15)(10,16)(11,14,12,13)$
$ 4, 4, 4, 4, 4, 2 $ $352$ $4$ $( 1, 2)( 3,22, 4,21)( 5,19, 6,20)( 7,17, 8,18)( 9,16,10,15)(11,14,12,13)$
$ 10, 10, 1, 1 $ $5632$ $10$ $( 3,17,20,13,10,22, 7, 6,12,16)( 4,18,19,14, 9,21, 8, 5,11,15)$
$ 20, 2 $ $5632$ $20$ $( 1, 2)( 3,18,20,14,10,21, 7, 5,12,15, 4,17,19,13, 9,22, 8, 6,11,16)$
$ 10, 10, 1, 1 $ $5632$ $10$ $( 3,13, 7,16,20,22,12,17,10, 6)( 4,14, 8,15,19,21,11,18, 9, 5)$
$ 20, 2 $ $5632$ $20$ $( 1, 2)( 3,14, 7,15,20,21,12,18,10, 5, 4,13, 8,16,19,22,11,17, 9, 6)$

Group invariants

Order:  $112640=2^{11} \cdot 5 \cdot 11$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.