Properties

Label 22T33
Order \(56320\)
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 $t$ :  $33$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,3,10,6,15)(2,4,9,5,16)(7,21,20,13,18,8,22,19,14,17)(11,12), (1,2)(3,12,7,9,19)(4,11,8,10,20)(5,22,14,18,15,6,21,13,17,16)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
5:  $C_5$
55:  $C_{11}:C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 11: $C_{11}:C_5$

Low degree siblings

22T33 x 2, 44T288

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $55$ $2$ $( 1, 2)( 7, 8)(11,12)(15,16)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $55$ $2$ $( 3, 4)(13,14)(15,16)(17,18)(19,20)(21,22)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $55$ $2$ $( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $55$ $2$ $( 5, 6)( 7, 8)(15,16)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $55$ $2$ $( 1, 2)( 3, 4)( 5, 6)(11,12)(13,14)(15,16)(19,20)(21,22)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $11$ $2$ $( 7, 8)(11,12)(13,14)(17,18)(19,20)(21,22)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $55$ $2$ $(13,14)(17,18)(19,20)(21,22)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $55$ $2$ $( 1, 2)( 7, 8)(11,12)(13,14)(15,16)(21,22)$
$ 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $55$ $2$ $( 1, 2)( 3, 4)( 7, 8)(11,12)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $55$ $2$ $( 9,10)(11,12)(15,16)(17,18)(19,20)(21,22)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $55$ $2$ $( 3, 4)( 9,10)(11,12)(13,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $55$ $2$ $( 1, 2)( 3, 4)( 7, 8)( 9,10)(13,14)(15,16)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $55$ $2$ $( 5, 6)( 7, 8)( 9,10)(11,12)(17,18)(21,22)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $55$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 9,10)(13,14)(17,18)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $55$ $2$ $( 1, 2)(15,16)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $11$ $2$ $( 1, 2)( 9,10)(11,12)(13,14)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $55$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)(13,14)(15,16)(19,20)(21,22)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $55$ $2$ $( 1, 2)( 7, 8)(11,12)(21,22)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $55$ $2$ $( 1, 2)( 7, 8)( 9,10)(13,14)(15,16)(21,22)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $11$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(19,20)(21,22)$
$ 11, 11 $ $5120$ $11$ $( 1,16, 7,22,14, 6,20,11, 3,17, 9)( 2,15, 8,21,13, 5,19,12, 4,18,10)$
$ 11, 11 $ $5120$ $11$ $( 1, 7,14,20, 3, 9,16,22, 6,11,17)( 2, 8,13,19, 4,10,15,21, 5,12,18)$
$ 5, 5, 5, 5, 1, 1 $ $2816$ $5$ $( 3,19, 9, 8,11)( 4,20,10, 7,12)( 5,16,17,13,21)( 6,15,18,14,22)$
$ 10, 5, 5, 2 $ $2816$ $10$ $( 1, 2)( 3,20,10, 8,12, 4,19, 9, 7,11)( 5,15,17,13,21)( 6,16,18,14,22)$
$ 10, 10, 1, 1 $ $2816$ $10$ $( 3,20,10, 8,12, 4,19, 9, 7,11)( 5,16,18,13,22, 6,15,17,14,21)$
$ 10, 5, 5, 2 $ $2816$ $10$ $( 1, 2)( 3,19, 9, 8,11)( 4,20,10, 7,12)( 5,15,18,13,22, 6,16,17,14,21)$
$ 5, 5, 5, 5, 1, 1 $ $2816$ $5$ $( 3, 9,11,19, 8)( 4,10,12,20, 7)( 5,17,21,16,13)( 6,18,22,15,14)$
$ 10, 5, 5, 2 $ $2816$ $10$ $( 1, 2)( 3, 9,12,19, 7, 4,10,11,20, 8)( 5,18,22,16,13)( 6,17,21,15,14)$
$ 10, 10, 1, 1 $ $2816$ $10$ $( 3, 9,12,19, 7, 4,10,11,20, 8)( 5,18,21,16,14, 6,17,22,15,13)$
$ 10, 5, 5, 2 $ $2816$ $10$ $( 1, 2)( 3, 9,11,19, 8)( 4,10,12,20, 7)( 5,17,22,16,14, 6,18,21,15,13)$
$ 5, 5, 5, 5, 1, 1 $ $2816$ $5$ $( 3,11, 8, 9,19)( 4,12, 7,10,20)( 5,21,13,17,16)( 6,22,14,18,15)$
$ 10, 5, 5, 2 $ $2816$ $10$ $( 1, 2)( 3,12, 8, 9,20, 4,11, 7,10,19)( 5,21,13,18,16)( 6,22,14,17,15)$
$ 10, 10, 1, 1 $ $2816$ $10$ $( 3,12, 8, 9,20, 4,11, 7,10,19)( 5,22,13,18,15, 6,21,14,17,16)$
$ 10, 5, 5, 2 $ $2816$ $10$ $( 1, 2)( 3,11, 8, 9,19)( 4,12, 7,10,20)( 5,22,13,17,15, 6,21,14,18,16)$
$ 5, 5, 5, 5, 1, 1 $ $2816$ $5$ $( 3, 8,19,11, 9)( 4, 7,20,12,10)( 5,13,16,21,17)( 6,14,15,22,18)$
$ 10, 5, 5, 2 $ $2816$ $10$ $( 1, 2)( 3, 7,19,12,10, 4, 8,20,11, 9)( 5,13,15,22,17)( 6,14,16,21,18)$
$ 10, 10, 1, 1 $ $2816$ $10$ $( 3, 7,19,12,10, 4, 8,20,11, 9)( 5,14,15,21,18, 6,13,16,22,17)$
$ 10, 5, 5, 2 $ $2816$ $10$ $( 1, 2)( 3, 8,19,11, 9)( 4, 7,20,12,10)( 5,14,16,22,18, 6,13,15,21,17)$

Group invariants

Order:  $56320=2^{10} \cdot 5 \cdot 11$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.