Properties

Label 22T41
Degree $22$
Order $887040$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no

Related objects

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Group action invariants

Degree $n$:  $22$
Transitive number $t$:  $41$
Parity:  $-1$
Primitive:  yes
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $1$
Generators:  (1,15,18,2,9,20,13,21,17,3,4,16)(5,12,22,11,8,14)(6,10,19,7), (1,22,17,12,15)(2,5,8,20,21)(3,16,11,9,7)(4,10,6,13,18)

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 11: None

Low degree siblings

44T405

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 $ $1$ $1$ $()$
$ 11, 11 $ $80640$ $11$ $( 1,12,18,14,11,22,13, 9, 6, 8, 2)( 3, 7,21,10,17,19,15, 5,16, 4,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $1155$ $2$ $( 1,22)( 2,19)( 4,13)( 5, 6)( 8,15)(10,21)(12,18)(17,20)$
$ 4, 4, 4, 4, 2, 2, 1, 1 $ $13860$ $4$ $( 1, 4,22,13)( 2,10,19,21)( 5,12, 6,18)( 7,16)( 8,20,15,17)( 9,14)$
$ 8, 8, 4, 1, 1 $ $55440$ $8$ $( 1, 6, 4,18,22, 5,13,12)( 2, 8,10,20,19,15,21,17)( 7, 9,16,14)$
$ 8, 8, 4, 2 $ $55440$ $8$ $( 1,21,20,19,22,10,17, 2)( 3,16,11, 7)( 4, 6,15,18,13, 5, 8,12)( 9,14)$
$ 4, 4, 4, 4, 2, 1, 1, 1, 1 $ $13860$ $4$ $( 1,15,22, 8)( 2,12,19,18)( 3,11)( 4,17,13,20)( 5,10, 6,21)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $12320$ $3$ $( 2,15,13)( 3, 9, 4)( 5,22,18)( 6,11,10)( 7,12,14)( 8,17,16)$
$ 4, 4, 4, 4, 2, 2, 2 $ $9240$ $4$ $( 1,19,21,20)( 2,18, 9, 7)( 3,14,13,22)( 4,12,15, 5)( 6,16)( 8,11)(10,17)$
$ 6, 6, 3, 3, 2, 2 $ $36960$ $6$ $( 1,21)( 2, 3,15, 9,13, 4)( 5, 7,22,12,18,14)( 6,10,11)( 8,16,17)(19,20)$
$ 12, 6, 4 $ $73920$ $12$ $( 1,20,21,19)( 2,12, 3,18,15,14, 9, 5,13, 7, 4,22)( 6, 8,10,16,11,17)$
$ 7, 7, 7, 1 $ $63360$ $7$ $( 1,19, 2,10, 4, 9, 5)( 3,22,13,16,14, 8,17)( 7,11,12,18,15,21,20)$
$ 7, 7, 7, 1 $ $63360$ $7$ $( 1, 5, 9, 4,10, 2,19)( 3,17, 8,14,16,13,22)( 7,20,21,15,18,12,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1386$ $2$ $( 1, 2)( 3, 5)( 4,14)( 6,15)( 7,17)( 8,21)( 9,11)(10,13)(12,19)(16,18)(20,22)$
$ 5, 5, 5, 5, 1, 1 $ $88704$ $5$ $( 1, 5,12,10,16)( 2, 3,19,13,18)( 4,11, 7, 6,22)( 9,17,15,20,14)$
$ 10, 10, 2 $ $88704$ $10$ $( 1,13, 5,18,12, 2,10, 3,16,19)( 4,15,11,20, 7,14, 6, 9,22,17)( 8,21)$
$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $330$ $2$ $( 1,14)( 2,17)( 3,10)( 4,22)( 5,16)( 8,19)( 9,13)$
$ 14, 7, 1 $ $63360$ $14$ $( 1, 3, 5,17, 9, 8, 4,14,10,16, 2,13,19,22)( 7,18,20,12,21,11,15)$
$ 14, 7, 1 $ $63360$ $14$ $( 1,17, 4,16,19, 3, 9,14, 2,22, 5, 8,10,13)( 7,12,15,20,11,18,21)$
$ 6, 6, 3, 3, 2, 1, 1 $ $73920$ $6$ $( 1, 5, 9,15, 3,12)( 2, 6,20)( 4, 7,16,10, 8,22)(11,13,19)(17,21)$
$ 4, 4, 4, 4, 2, 2, 1, 1 $ $27720$ $4$ $( 1,15, 4,20)( 2,21,18, 5)( 3,16)( 6,19,10,12)( 7,11)( 8,13,17,22)$

Group invariants

Order:  $887040=2^{8} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11$
Cyclic:  no
Abelian:  no
Solvable:  no
GAP id:  not available
Character table: not available.