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Properties

Label 13T1
Order \(13\)
n \(13\)
Cyclic Yes
Abelian Yes
Solvable Yes
Primitive Yes
$p$-group Yes
Group: $C_{13}$

Related objects

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Group action invariants

Degree $n$ :  $13$
Transitive number $t$ :  $1$
Group :  $C_{13}$
CHM label :  $C(13)=13$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $1$
Generators:  (1,2,3,4,5,6,7,8,9,10,11,12,13)
$|\Aut(F/K)|$:  $13$

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 $ $1$ $1$ $()$
$ 13 $ $1$ $13$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13)$
$ 13 $ $1$ $13$ $( 1, 3, 5, 7, 9,11,13, 2, 4, 6, 8,10,12)$
$ 13 $ $1$ $13$ $( 1, 4, 7,10,13, 3, 6, 9,12, 2, 5, 8,11)$
$ 13 $ $1$ $13$ $( 1, 5, 9,13, 4, 8,12, 3, 7,11, 2, 6,10)$
$ 13 $ $1$ $13$ $( 1, 6,11, 3, 8,13, 5,10, 2, 7,12, 4, 9)$
$ 13 $ $1$ $13$ $( 1, 7,13, 6,12, 5,11, 4,10, 3, 9, 2, 8)$
$ 13 $ $1$ $13$ $( 1, 8, 2, 9, 3,10, 4,11, 5,12, 6,13, 7)$
$ 13 $ $1$ $13$ $( 1, 9, 4,12, 7, 2,10, 5,13, 8, 3,11, 6)$
$ 13 $ $1$ $13$ $( 1,10, 6, 2,11, 7, 3,12, 8, 4,13, 9, 5)$
$ 13 $ $1$ $13$ $( 1,11, 8, 5, 2,12, 9, 6, 3,13,10, 7, 4)$
$ 13 $ $1$ $13$ $( 1,12,10, 8, 6, 4, 2,13,11, 9, 7, 5, 3)$
$ 13 $ $1$ $13$ $( 1,13,12,11,10, 9, 8, 7, 6, 5, 4, 3, 2)$

Group invariants

Order:  $13$ (is prime)
Cyclic:  Yes
Abelian:  Yes
Solvable:  Yes
GAP id:  [13, 1]
Character table:    
     13  1   1   1   1   1   1   1   1   1   1   1   1   1

        1a 13a 13b 13c 13d 13e 13f 13g 13h 13i 13j 13k 13l

X.1      1   1   1   1   1   1   1   1   1   1   1   1   1
X.2      1   A   B   C   D   E   F  /F  /E  /D  /C  /B  /A
X.3      1   B   D   F  /E  /C  /A   A   C   E  /F  /D  /B
X.4      1   C   F  /D  /A   B   E  /E  /B   A   D  /F  /C
X.5      1   D  /E  /A   C  /F  /B   B   F  /C   A   E  /D
X.6      1   E  /C   B  /F  /A   D  /D   A   F  /B   C  /E
X.7      1   F  /A   E  /B   D  /C   C  /D   B  /E   A  /F
X.8      1  /F   A  /E   B  /D   C  /C   D  /B   E  /A   F
X.9      1  /E   C  /B   F   A  /D   D  /A  /F   B  /C   E
X.10     1  /D   E   A  /C   F   B  /B  /F   C  /A  /E   D
X.11     1  /C  /F   D   A  /B  /E   E   B  /A  /D   F   C
X.12     1  /B  /D  /F   E   C   A  /A  /C  /E   F   D   B
X.13     1  /A  /B  /C  /D  /E  /F   F   E   D   C   B   A

A = E(13)
B = E(13)^2
C = E(13)^3
D = E(13)^4
E = E(13)^5
F = E(13)^6