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Properties

Label 22T13
Degree $22$
Order $1320$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $C_2\times \PSL(2,11)$

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Show commands: Magma

magma: G := TransitiveGroup(22, 13);
 

Group action invariants

Degree $n$:  $22$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $13$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\times \PSL(2,11)$
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:  (1,20,12,5,22,2,19,11,6,21)(3,17,13,8,16,4,18,14,7,15)(9,10), (1,21,15,3,8,10)(2,22,16,4,7,9)(5,14,11,6,13,12)(17,20)(18,19)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$660$:  $\PSL(2,11)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 11: $\PSL(2,11)$

Low degree siblings

22T13, 24T2948

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

LabelCycle 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, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)$
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ $110$ $3$ $( 3,13,10)( 4,14, 9)( 5,21,11)( 6,22,12)( 7,18,16)( 8,17,15)$
$ 6, 6, 6, 2, 2 $ $110$ $6$ $( 1, 2)( 3,14,10, 4,13, 9)( 5,22,11, 6,21,12)( 7,17,16, 8,18,15)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $55$ $2$ $( 3,10)( 4, 9)( 5,16)( 6,15)( 7,11)( 8,12)(17,22)(18,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $55$ $2$ $( 1, 2)( 3, 9)( 4,10)( 5,15)( 6,16)( 7,12)( 8,11)(13,14)(17,21)(18,22)(19,20)$
$ 10, 10, 2 $ $132$ $10$ $( 1, 2)( 3,17, 5, 8,11, 4,18, 6, 7,12)( 9,16,19,21,14,10,15,20,22,13)$
$ 5, 5, 5, 5, 1, 1 $ $132$ $5$ $( 3,18, 5, 7,11)( 4,17, 6, 8,12)( 9,15,19,22,14)(10,16,20,21,13)$
$ 10, 10, 2 $ $132$ $10$ $( 1, 2)( 3, 9,18,19,21, 4,10,17,20,22)( 5,14,16,12, 7, 6,13,15,11, 8)$
$ 5, 5, 5, 5, 1, 1 $ $132$ $5$ $( 3,10,18,20,21)( 4, 9,17,19,22)( 5,13,16,11, 7)( 6,14,15,12, 8)$
$ 6, 6, 6, 2, 2 $ $110$ $6$ $( 1,20)( 2,19)( 3, 9,13, 4,10,14)( 5, 8,21,17,11,15)( 6, 7,22,18,12,16)$
$ 6, 6, 3, 3, 2, 2 $ $110$ $6$ $( 1,19)( 2,20)( 3,10,13)( 4, 9,14)( 5, 7,21,18,11,16)( 6, 8,22,17,12,15)$
$ 22 $ $60$ $22$ $( 1,20,22,18, 6, 3,15,11, 9,13, 8, 2,19,21,17, 5, 4,16,12,10,14, 7)$
$ 11, 11 $ $60$ $11$ $( 1,19,22,17, 6, 4,15,12, 9,14, 8)( 2,20,21,18, 5, 3,16,11,10,13, 7)$
$ 22 $ $60$ $22$ $( 1,20,22, 3,14,16, 6,10, 8,11,17, 2,19,21, 4,13,15, 5, 9, 7,12,18)$
$ 11, 11 $ $60$ $11$ $( 1,19,22, 4,14,15, 6, 9, 8,12,17)( 2,20,21, 3,13,16, 5,10, 7,11,18)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $1320=2^{3} \cdot 3 \cdot 5 \cdot 11$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  1320.134
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 5A1 5A2 6A 6B 6C 10A1 10A3 11A1 11A-1 22A1 22A-1
Size 1 1 55 55 110 132 132 110 110 110 132 132 60 60 60 60
2 P 1A 1A 1A 1A 3A 5A2 5A1 3A 3A 3A 5A1 5A2 11A-1 11A1 11A1 11A-1
3 P 1A 2A 2B 2C 1A 5A2 5A1 2C 2B 2A 10A3 10A1 11A1 11A-1 22A1 22A-1
5 P 1A 2A 2B 2C 3A 1A 1A 6C 6A 6B 2A 2A 11A1 11A-1 22A1 22A-1
11 P 1A 2A 2B 2C 3A 5A1 5A2 6C 6A 6B 10A1 10A3 1A 1A 2A 2A
Type
1320.134.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1320.134.1b R 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1
1320.134.5a1 C 5 5 1 1 −1 0 0 1 −1 1 0 0 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115
1320.134.5a2 C 5 5 1 1 −1 0 0 1 −1 1 0 0 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115
1320.134.5b1 C 5 −5 1 −1 −1 0 0 1 1 −1 0 0 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−ζ11−ζ113−ζ114−ζ115 ζ11−2+1+ζ11+ζ113+ζ114+ζ115
1320.134.5b2 C 5 −5 1 −1 −1 0 0 1 1 −1 0 0 ζ11−2+ζ11+ζ113+ζ114+ζ115 −ζ11−2−1−ζ11−ζ113−ζ114−ζ115 ζ11−2+1+ζ11+ζ113+ζ114+ζ115 −ζ11−2−ζ11−ζ113−ζ114−ζ115
1320.134.10a R 10 10 2 2 1 0 0 −1 1 −1 0 0 −1 −1 −1 −1
1320.134.10b R 10 −10 2 −2 1 0 0 −1 −1 1 0 0 −1 −1 1 1
1320.134.10c R 10 10 −2 −2 1 0 0 1 1 1 0 0 −1 −1 −1 −1
1320.134.10d R 10 −10 −2 2 1 0 0 1 −1 −1 0 0 −1 −1 1 1
1320.134.11a R 11 11 −1 −1 −1 1 1 −1 −1 −1 1 1 0 0 0 0
1320.134.11b R 11 −11 −1 1 −1 1 1 −1 1 1 −1 −1 0 0 0 0
1320.134.12a1 R 12 12 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 1 1 1 1
1320.134.12a2 R 12 12 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 1 1 1 1
1320.134.12b1 R 12 −12 0 0 0 ζ5−2+ζ52 ζ5−1+ζ5 0 0 0 −ζ5−1−ζ5 −ζ5−2−ζ52 1 1 −1 −1
1320.134.12b2 R 12 −12 0 0 0 ζ5−1+ζ5 ζ5−2+ζ52 0 0 0 −ζ5−2−ζ52 −ζ5−1−ζ5 1 1 −1 −1

magma: CharacterTable(G);