↑

Properties

Label 22T14
Degree $22$
Order $1320$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $\PGL(2,11)$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(22, 14);
 

Group action invariants

Degree $n$:  $22$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $14$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\PGL(2,11)$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,14,8,18,2,16,5,21,3,15)(4,19,6,17,9,22,7,12,11,13)(10,20), (1,3,9)(2,4,10,11,7,5)(6,8)(12,14,15,19,13,20)(16,17,21)(18,22)
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: $C_2$

Degree 11: None

Low degree siblings

12T218, 24T2949

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$1^{22}$ $1$ $1$ $()$
$2^{11}$ $66$ $2$ $( 1,17)( 2,13)( 3,19)( 4,14)( 5,18)( 6,16)( 7,20)( 8,21)( 9,12)(10,22)(11,15)$
$5^{4},1^{2}$ $132$ $5$ $( 1,11, 3, 9, 4)( 2,10, 7, 6, 8)(12,14,17,15,19)(13,22,20,16,21)$
$5^{4},1^{2}$ $132$ $5$ $( 1, 9,11, 4, 3)( 2, 6,10, 8, 7)(12,15,14,19,17)(13,16,22,21,20)$
$10^{2},2$ $132$ $10$ $( 1,12,11,14, 3,17, 9,15, 4,19)( 2,16,10,21, 7,13, 6,22, 8,20)( 5,18)$
$10^{2},2$ $132$ $10$ $( 1,15, 3,12, 4,17,11,19, 9,14)( 2,22, 7,16, 8,13,10,20, 6,21)( 5,18)$
$11^{2}$ $120$ $11$ $( 1,11,10, 2, 6, 9, 7, 3, 5, 8, 4)(12,13,19,21,20,22,14,17,15,16,18)$
$2^{8},1^{6}$ $55$ $2$ $( 2, 4)( 3, 9)( 5,10)( 7,11)(13,15)(14,20)(16,21)(18,22)$
$4^{4},2^{3}$ $110$ $4$ $( 1,17)( 2,15, 4,13)( 3,22, 9,18)( 5,16,10,21)( 6,12)( 7,20,11,14)( 8,19)$
$3^{6},1^{4}$ $110$ $3$ $( 1, 6, 8)( 2, 9,11)( 3, 7, 4)(12,19,17)(13,22,20)(14,15,18)$
$6^{2},3^{2},2^{2}$ $110$ $6$ $( 1, 8, 6)( 2, 7, 9, 4,11, 3)( 5,10)(12,17,19)(13,14,22,15,20,18)(16,21)$
$12,6,4$ $110$ $12$ $( 1,12, 8,17, 6,19)( 2,18, 7,13, 9,14, 4,22,11,15, 3,20)( 5,16,10,21)$
$12,6,4$ $110$ $12$ $( 1,12, 8,17, 6,19)( 2,22, 7,15, 9,20, 4,18,11,13, 3,14)( 5,21,10,16)$

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:  1320.133
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 3A 4A 5A1 5A2 6A 10A1 10A3 11A 12A1 12A5
Size 1 55 66 110 110 132 132 110 132 132 120 110 110
2 P 1A 1A 1A 3A 2A 5A2 5A1 3A 5A1 5A2 11A 6A 6A
3 P 1A 2A 2B 1A 4A 5A2 5A1 2A 10A3 10A1 11A 4A 4A
5 P 1A 2A 2B 3A 4A 1A 1A 6A 2B 2B 11A 12A5 12A1
11 P 1A 2A 2B 3A 4A 5A1 5A2 6A 10A1 10A3 1A 12A1 12A5
Type
1320.133.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
1320.133.1b R 1 1 −1 1 −1 1 1 1 −1 −1 1 −1 −1
1320.133.10a R 10 −2 0 1 2 0 0 1 0 0 −1 −1 −1
1320.133.10b R 10 2 0 −2 0 0 0 2 0 0 −1 0 0
1320.133.10c R 10 −2 0 1 −2 0 0 1 0 0 −1 1 1
1320.133.10d1 R 10 2 0 1 0 0 0 −1 0 0 −1 −ζ12−1−ζ12 ζ12−1+ζ12
1320.133.10d2 R 10 2 0 1 0 0 0 −1 0 0 −1 ζ12−1+ζ12 −ζ12−1−ζ12
1320.133.11a R 11 −1 1 −1 −1 1 1 −1 1 1 0 −1 −1
1320.133.11b R 11 −1 −1 −1 1 1 1 −1 −1 −1 0 1 1
1320.133.12a1 R 12 0 2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 0 ζ5−1+ζ5 ζ5−2+ζ52 1 0 0
1320.133.12a2 R 12 0 2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 0 ζ5−2+ζ52 ζ5−1+ζ5 1 0 0
1320.133.12b1 R 12 0 −2 0 0 ζ5−2+ζ52 ζ5−1+ζ5 0 −ζ5−1−ζ5 −ζ5−2−ζ52 1 0 0
1320.133.12b2 R 12 0 −2 0 0 ζ5−1+ζ5 ζ5−2+ζ52 0 −ζ5−2−ζ52 −ζ5−1−ζ5 1 0 0

magma: CharacterTable(G);