# Properties

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

# Related objects

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 Type Size Order Representative $1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 1$ $7$ $3$ $(2,3,5)(4,7,6)$ $3, 3, 1$ $7$ $3$ $(2,5,3)(4,6,7)$ $7$ $3$ $7$ $(1,2,3,4,5,6,7)$ $7$ $3$ $7$ $(1,4,7,3,6,2,5)$

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);