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Properties

Label 7T1
Degree $7$
Order $7$
Cyclic yes
Abelian yes
Solvable yes
Primitive yes
$p$-group yes
Group: $C_7$

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Show commands: Magma

magma: G := TransitiveGroup(7, 1);
 

Group action invariants

Degree $n$:  $7$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_7$
CHM label:  $C(7) = 7$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
Nilpotency class:  $1$
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $7$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2,3,4,5,6,7)
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 $ $1$ $1$ $()$
$ 7 $ $1$ $7$ $(1,2,3,4,5,6,7)$
$ 7 $ $1$ $7$ $(1,3,5,7,2,4,6)$
$ 7 $ $1$ $7$ $(1,4,7,3,6,2,5)$
$ 7 $ $1$ $7$ $(1,5,2,6,3,7,4)$
$ 7 $ $1$ $7$ $(1,6,4,2,7,5,3)$
$ 7 $ $1$ $7$ $(1,7,6,5,4,3,2)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $7$ (is prime)
magma: Order(G);
 
Cyclic:  yes
magma: IsCyclic(G);
 
Abelian:  yes
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Label:  7.1
magma: IdentifyGroup(G);
 
Character table:    
     7  1  1  1  1  1  1  1

       1a 7a 7b 7c 7d 7e 7f

X.1     1  1  1  1  1  1  1
X.2     1  A  B  C /C /B /A
X.3     1  B /C /A  A  C /B
X.4     1  C /A  B /B  A /C
X.5     1 /C  A /B  B /A  C
X.6     1 /B  C  A /A /C  B
X.7     1 /A /B /C  C  B  A

A = E(7)
B = E(7)^2
C = E(7)^3

magma: CharacterTable(G);