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Properties

Label 22T3
Degree $22$
Order $44$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{22}$

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Show commands: Magma

magma: G := TransitiveGroup(22, 3);
 

Group action invariants

Degree $n$:  $22$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $3$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_{22}$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,19)(2,20)(3,17)(4,18)(5,15)(6,16)(7,14)(8,13)(9,11)(10,12), (1,21)(2,22)(3,19)(4,20)(5,18)(6,17)(7,15)(8,16)(9,14)(10,13)(11,12)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$22$:  $D_{11}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: $D_{11}$

Low degree siblings

22T3, 44T4

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$1^{22}$ $1$ $1$ $()$
$2^{10},1^{2}$ $11$ $2$ $( 3,21)( 4,22)( 5,20)( 6,19)( 7,18)( 8,17)( 9,15)(10,16)(11,13)(12,14)$
$2^{11}$ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)$
$2^{11}$ $11$ $2$ $( 1, 2)( 3,22)( 4,21)( 5,19)( 6,20)( 7,17)( 8,18)( 9,16)(10,15)(11,14)(12,13)$
$22$ $2$ $22$ $( 1, 3, 6, 8,10,11,14,15,18,20,22, 2, 4, 5, 7, 9,12,13,16,17,19,21)$
$11^{2}$ $2$ $11$ $( 1, 4, 6, 7,10,12,14,16,18,19,22)( 2, 3, 5, 8, 9,11,13,15,17,20,21)$
$22$ $2$ $22$ $( 1, 5,10,13,18,21, 4, 8,12,15,19, 2, 6, 9,14,17,22, 3, 7,11,16,20)$
$11^{2}$ $2$ $11$ $( 1, 6,10,14,18,22, 4, 7,12,16,19)( 2, 5, 9,13,17,21, 3, 8,11,15,20)$
$11^{2}$ $2$ $11$ $( 1, 7,14,19, 4,10,16,22, 6,12,18)( 2, 8,13,20, 3, 9,15,21, 5,11,17)$
$22$ $2$ $22$ $( 1, 8,14,20, 4, 9,16,21, 6,11,18, 2, 7,13,19, 3,10,15,22, 5,12,17)$
$22$ $2$ $22$ $( 1, 9,18, 3,12,20, 6,13,22, 8,16, 2,10,17, 4,11,19, 5,14,21, 7,15)$
$11^{2}$ $2$ $11$ $( 1,10,18, 4,12,19, 6,14,22, 7,16)( 2, 9,17, 3,11,20, 5,13,21, 8,15)$
$22$ $2$ $22$ $( 1,11,22, 9,19, 8,18, 5,16, 3,14, 2,12,21,10,20, 7,17, 6,15, 4,13)$
$11^{2}$ $2$ $11$ $( 1,12,22,10,19, 7,18, 6,16, 4,14)( 2,11,21, 9,20, 8,17, 5,15, 3,13)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $44=2^{2} \cdot 11$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  44.3
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 11A1 11A2 11A3 11A4 11A5 22A1 22A3 22A5 22A7 22A9
Size 1 1 11 11 2 2 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 11A4 11A5 11A2 11A3 11A1 11A4 11A5 11A1 11A2 11A3
11 P 1A 2A 2B 2C 11A1 11A4 11A5 11A2 11A3 22A9 22A3 22A5 22A1 22A7
Type
44.3.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
44.3.1b R 1 −1 −1 1 1 1 1 1 1 −1 −1 −1 −1 −1
44.3.1c R 1 −1 1 −1 1 1 1 1 1 −1 −1 −1 −1 −1
44.3.1d R 1 1 −1 −1 1 1 1 1 1 1 1 1 1 1
44.3.2a1 R 2 2 0 0 ζ11−5+ζ115 ζ11−1+ζ11 ζ11−4+ζ114 ζ11−2+ζ112 ζ11−3+ζ113 ζ11−3+ζ113 ζ11−2+ζ112 ζ11−4+ζ114 ζ11−1+ζ11 ζ11−5+ζ115
44.3.2a2 R 2 2 0 0 ζ11−4+ζ114 ζ11−3+ζ113 ζ11−1+ζ11 ζ11−5+ζ115 ζ11−2+ζ112 ζ11−2+ζ112 ζ11−5+ζ115 ζ11−1+ζ11 ζ11−3+ζ113 ζ11−4+ζ114
44.3.2a3 R 2 2 0 0 ζ11−3+ζ113 ζ11−5+ζ115 ζ11−2+ζ112 ζ11−1+ζ11 ζ11−4+ζ114 ζ11−4+ζ114 ζ11−1+ζ11 ζ11−2+ζ112 ζ11−5+ζ115 ζ11−3+ζ113
44.3.2a4 R 2 2 0 0 ζ11−2+ζ112 ζ11−4+ζ114 ζ11−5+ζ115 ζ11−3+ζ113 ζ11−1+ζ11 ζ11−1+ζ11 ζ11−3+ζ113 ζ11−5+ζ115 ζ11−4+ζ114 ζ11−2+ζ112
44.3.2a5 R 2 2 0 0 ζ11−1+ζ11 ζ11−2+ζ112 ζ11−3+ζ113 ζ11−4+ζ114 ζ11−5+ζ115 ζ11−5+ζ115 ζ11−4+ζ114 ζ11−3+ζ113 ζ11−2+ζ112 ζ11−1+ζ11
44.3.2b1 R 2 −2 0 0 ζ11−5+ζ115 ζ11−1+ζ11 ζ11−4+ζ114 ζ11−2+ζ112 ζ11−3+ζ113 −ζ11−3−ζ113 −ζ11−2−ζ112 −ζ11−4−ζ114 −ζ11−1−ζ11 −ζ11−5−ζ115
44.3.2b2 R 2 −2 0 0 ζ11−4+ζ114 ζ11−3+ζ113 ζ11−1+ζ11 ζ11−5+ζ115 ζ11−2+ζ112 −ζ11−2−ζ112 −ζ11−5−ζ115 −ζ11−1−ζ11 −ζ11−3−ζ113 −ζ11−4−ζ114
44.3.2b3 R 2 −2 0 0 ζ11−3+ζ113 ζ11−5+ζ115 ζ11−2+ζ112 ζ11−1+ζ11 ζ11−4+ζ114 −ζ11−4−ζ114 −ζ11−1−ζ11 −ζ11−2−ζ112 −ζ11−5−ζ115 −ζ11−3−ζ113
44.3.2b4 R 2 −2 0 0 ζ11−2+ζ112 ζ11−4+ζ114 ζ11−5+ζ115 ζ11−3+ζ113 ζ11−1+ζ11 −ζ11−1−ζ11 −ζ11−3−ζ113 −ζ11−5−ζ115 −ζ11−4−ζ114 −ζ11−2−ζ112
44.3.2b5 R 2 −2 0 0 ζ11−1+ζ11 ζ11−2+ζ112 ζ11−3+ζ113 ζ11−4+ζ114 ζ11−5+ζ115 −ζ11−5−ζ115 −ζ11−4−ζ114 −ζ11−3−ζ113 −ζ11−2−ζ112 −ζ11−1−ζ11

magma: CharacterTable(G);