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Properties

Label 7T3
Degree $7$
Order $21$
Cyclic no
Abelian no
Solvable yes
Primitive yes
$p$-group no
Group: $C_7:C_3$

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Show commands: Magma

magma: G := TransitiveGroup(7, 3);
 

Group action invariants

Degree $n$:  $7$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $3$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_7:C_3$
CHM label:  $F_{21}(7) = 7:3$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
Nilpotency class:  $-1$ (not nilpotent)
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,2,4)(3,6,5), (1,2,3,4,5,6,7)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$3$:  $C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

21T2

Siblings are shown 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 $ $1$ $1$ $()$
$ 3, 3, 1 $ $7$ $3$ $(2,3,5)(4,7,6)$
$ 3, 3, 1 $ $7$ $3$ $(2,5,3)(4,6,7)$
$ 7 $ $3$ $7$ $(1,2,3,4,5,6,7)$
$ 7 $ $3$ $7$ $(1,4,7,3,6,2,5)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $21=3 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Label:  21.1
magma: IdentifyGroup(G);
 
Character table:    
     3  1  1  1  .  .
     7  1  .  .  1  1

       1a 3a 3b 7a 7b
    2P 1a 3b 3a 7a 7b
    3P 1a 1a 1a 7b 7a
    5P 1a 3b 3a 7b 7a
    7P 1a 3a 3b 1a 1a

X.1     1  1  1  1  1
X.2     1  A /A  1  1
X.3     1 /A  A  1  1
X.4     3  .  .  B /B
X.5     3  .  . /B  B

A = E(3)^2
  = (-1-Sqrt(-3))/2 = -1-b3
B = E(7)+E(7)^2+E(7)^4
  = (-1+Sqrt(-7))/2 = b7

magma: CharacterTable(G);