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Properties

Label 19T1
Degree $19$
Order $19$
Cyclic yes
Abelian yes
Solvable yes
Primitive yes
$p$-group yes
Group: $C_{19}$

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Show commands: Magma

magma: G := TransitiveGroup(19, 1);
 

Group action invariants

Degree $n$:  $19$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{19}$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $19$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19)
magma: Generators(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

There are no siblings 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 $ $1$ $1$ $()$
$ 19 $ $1$ $19$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15,16,17,18,19)$
$ 19 $ $1$ $19$ $( 1, 3, 5, 7, 9,11,13,15,17,19, 2, 4, 6, 8,10,12,14,16,18)$
$ 19 $ $1$ $19$ $( 1, 4, 7,10,13,16,19, 3, 6, 9,12,15,18, 2, 5, 8,11,14,17)$
$ 19 $ $1$ $19$ $( 1, 5, 9,13,17, 2, 6,10,14,18, 3, 7,11,15,19, 4, 8,12,16)$
$ 19 $ $1$ $19$ $( 1, 6,11,16, 2, 7,12,17, 3, 8,13,18, 4, 9,14,19, 5,10,15)$
$ 19 $ $1$ $19$ $( 1, 7,13,19, 6,12,18, 5,11,17, 4,10,16, 3, 9,15, 2, 8,14)$
$ 19 $ $1$ $19$ $( 1, 8,15, 3,10,17, 5,12,19, 7,14, 2, 9,16, 4,11,18, 6,13)$
$ 19 $ $1$ $19$ $( 1, 9,17, 6,14, 3,11,19, 8,16, 5,13, 2,10,18, 7,15, 4,12)$
$ 19 $ $1$ $19$ $( 1,10,19, 9,18, 8,17, 7,16, 6,15, 5,14, 4,13, 3,12, 2,11)$
$ 19 $ $1$ $19$ $( 1,11, 2,12, 3,13, 4,14, 5,15, 6,16, 7,17, 8,18, 9,19,10)$
$ 19 $ $1$ $19$ $( 1,12, 4,15, 7,18,10, 2,13, 5,16, 8,19,11, 3,14, 6,17, 9)$
$ 19 $ $1$ $19$ $( 1,13, 6,18,11, 4,16, 9, 2,14, 7,19,12, 5,17,10, 3,15, 8)$
$ 19 $ $1$ $19$ $( 1,14, 8, 2,15, 9, 3,16,10, 4,17,11, 5,18,12, 6,19,13, 7)$
$ 19 $ $1$ $19$ $( 1,15,10, 5,19,14, 9, 4,18,13, 8, 3,17,12, 7, 2,16,11, 6)$
$ 19 $ $1$ $19$ $( 1,16,12, 8, 4,19,15,11, 7, 3,18,14,10, 6, 2,17,13, 9, 5)$
$ 19 $ $1$ $19$ $( 1,17,14,11, 8, 5, 2,18,15,12, 9, 6, 3,19,16,13,10, 7, 4)$
$ 19 $ $1$ $19$ $( 1,18,16,14,12,10, 8, 6, 4, 2,19,17,15,13,11, 9, 7, 5, 3)$
$ 19 $ $1$ $19$ $( 1,19,18,17,16,15,14,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $19$ (is prime)
magma: Order(G);
 
Cyclic:  yes
magma: IsCyclic(G);
 
Abelian:  yes
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $1$
Label:  19.1
magma: IdentifyGroup(G);
 
Character table:    
     19  1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1

        1a 19a 19b 19c 19d 19e 19f 19g 19h 19i 19j 19k 19l 19m 19n 19o 19p 19q

X.1      1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1   1
X.2      1   A   B   C   D   E   F   G   H   I  /I  /H  /G  /F  /E  /D  /C  /B
X.3      1   B   D   F   H  /I  /G  /E  /C  /A   A   C   E   G   I  /H  /F  /D
X.4      1   C   F   I  /G  /D  /A   B   E   H  /H  /E  /B   A   D   G  /I  /F
X.5      1   D   H  /G  /C   A   E   I  /F  /B   B   F  /I  /E  /A   C   G  /H
X.6      1   E  /I  /D   A   F  /H  /C   B   G  /G  /B   C   H  /F  /A   D   I
X.7      1   F  /G  /A   E  /H  /B   D  /I  /C   C   I  /D   B   H  /E   A   G
X.8      1   G  /E   B   I  /C   D  /H  /A   F  /F   A   H  /D   C  /I  /B   E
X.9      1   H  /C   E  /F   B  /I  /A   G  /D   D  /G   A   I  /B   F  /E   C
X.10     1   I  /A   H  /B   G  /C   F  /D   E  /E   D  /F   C  /G   B  /H   A
X.11     1  /I   A  /H   B  /G   C  /F   D  /E   E  /D   F  /C   G  /B   H  /A
X.12     1  /H   C  /E   F  /B   I   A  /G   D  /D   G  /A  /I   B  /F   E  /C
X.13     1  /G   E  /B  /I   C  /D   H   A  /F   F  /A  /H   D  /C   I   B  /E
X.14     1  /F   G   A  /E   H   B  /D   I   C  /C  /I   D  /B  /H   E  /A  /G
X.15     1  /E   I   D  /A  /F   H   C  /B  /G   G   B  /C  /H   F   A  /D  /I
X.16     1  /D  /H   G   C  /A  /E  /I   F   B  /B  /F   I   E   A  /C  /G   H
X.17     1  /C  /F  /I   G   D   A  /B  /E  /H   H   E   B  /A  /D  /G   I   F
X.18     1  /B  /D  /F  /H   I   G   E   C   A  /A  /C  /E  /G  /I   H   F   D
X.19     1  /A  /B  /C  /D  /E  /F  /G  /H  /I   I   H   G   F   E   D   C   B

     19   1

        19r

X.1       1
X.2      /A
X.3      /B
X.4      /C
X.5      /D
X.6      /E
X.7      /F
X.8      /G
X.9      /H
X.10     /I
X.11      I
X.12      H
X.13      G
X.14      F
X.15      E
X.16      D
X.17      C
X.18      B
X.19      A

A = E(19)
B = E(19)^2
C = E(19)^3
D = E(19)^4
E = E(19)^5
F = E(19)^6
G = E(19)^7
H = E(19)^8
I = E(19)^9

magma: CharacterTable(G);