# Properties

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

# Related objects

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); 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

 Label Cycle Type Size Order Index Representative 1A $1^{7}$ $1$ $1$ $0$ $()$ 7A1 $7$ $1$ $7$ $6$ $(1,3,5,7,2,4,6)$ 7A-1 $7$ $1$ $7$ $6$ $(1,4,7,3,6,2,5)$ 7A2 $7$ $1$ $7$ $6$ $(1,6,4,2,7,5,3)$ 7A-2 $7$ $1$ $7$ $6$ $(1,2,3,4,5,6,7)$ 7A3 $7$ $1$ $7$ $6$ $(1,5,2,6,3,7,4)$ 7A-3 $7$ $1$ $7$ $6$ $(1,7,6,5,4,3,2)$

magma: ConjugacyClasses(G);

Malle's constant $a(G)$:     $1/6$

## Group invariants

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

 1A 7A1 7A-1 7A2 7A-2 7A3 7A-3 Size 1 1 1 1 1 1 1 7 P 1A 7A-3 7A-1 7A3 7A2 7A1 7A-2 Type 7.1.1a R $1$ $1$ $1$ $1$ $1$ $1$ $1$ 7.1.1b1 C $1$ $ζ7−3$ $ζ73$ $ζ7$ $ζ7−1$ $ζ7−2$ $ζ72$ 7.1.1b2 C $1$ $ζ73$ $ζ7−3$ $ζ7−1$ $ζ7$ $ζ72$ $ζ7−2$ 7.1.1b3 C $1$ $ζ7−2$ $ζ72$ $ζ73$ $ζ7−3$ $ζ7$ $ζ7−1$ 7.1.1b4 C $1$ $ζ72$ $ζ7−2$ $ζ7−3$ $ζ73$ $ζ7−1$ $ζ7$ 7.1.1b5 C $1$ $ζ7−1$ $ζ7$ $ζ7−2$ $ζ72$ $ζ7−3$ $ζ73$ 7.1.1b6 C $1$ $ζ7$ $ζ7−1$ $ζ72$ $ζ7−2$ $ζ73$ $ζ7−3$

magma: CharacterTable(G);