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Properties

Label 22T41
Degree $22$
Order $887040$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $M_{22}:C_2$

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Show commands: Magma

magma: G := TransitiveGroup(22, 41);
 

Group action invariants

Degree $n$:  $22$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $41$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $M_{22}:C_2$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
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)
magma: Generators(G);
 

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $887040=2^{8} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  887040.a
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);