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Properties

Label 22T4
Degree $22$
Order $110$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $F_{11}$

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Show commands: Magma

magma: G := TransitiveGroup(22, 4);
 

Group action invariants

Degree $n$:  $22$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $4$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $F_{11}$
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:  (1,3,8,16,10,19,17,14,5,11)(2,4,7,15,9,20,18,13,6,12)(21,22), (1,4,5,8,10,12,13,15,17,20,21)(2,3,6,7,9,11,14,16,18,19,22)
magma: Generators(G);
 

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: $F_{11}$

Low degree siblings

11T4

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $110=2 \cdot 5 \cdot 11$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  110.1
magma: IdentifyGroup(G);
 
Character table:

1A 2A 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 11A
Size 1 11 11 11 11 11 11 11 11 11 10
2 P 1A 1A 5A1 5A-1 5A-2 5A2 5A1 5A2 5A-1 5A-2 11A
5 P 1A 2A 1A 1A 1A 1A 2A 2A 2A 2A 11A
11 P 1A 2A 5A-2 5A2 5A-1 5A1 10A1 10A-3 10A-1 10A3 1A
Type
110.1.1a R 1 1 1 1 1 1 1 1 1 1 1
110.1.1b R 1 −1 1 1 1 1 −1 −1 −1 −1 1
110.1.1c1 C 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 1
110.1.1c2 C 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52 1
110.1.1c3 C 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1 1
110.1.1c4 C 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 1
110.1.1d1 C 1 −1 ζ5−2 ζ52 ζ5 ζ5−1 −ζ5−1 −ζ5 −ζ52 −ζ5−2 1
110.1.1d2 C 1 −1 ζ52 ζ5−2 ζ5−1 ζ5 −ζ5 −ζ5−1 −ζ5−2 −ζ52 1
110.1.1d3 C 1 −1 ζ5−1 ζ5 ζ5−2 ζ52 −ζ52 −ζ5−2 −ζ5 −ζ5−1 1
110.1.1d4 C 1 −1 ζ5 ζ5−1 ζ52 ζ5−2 −ζ5−2 −ζ52 −ζ5−1 −ζ5 1
110.1.10a R 10 0 0 0 0 0 0 0 0 0 −1

magma: CharacterTable(G);