# Properties

 Label 7T4 Degree $7$ Order $42$ Cyclic no Abelian no Solvable yes Primitive yes $p$-group no Group: $F_7$

# Related objects

## Group action invariants

 Degree $n$: $7$ Transitive number $t$: $4$ Group: $F_7$ CHM label: $F_{42}(7) = 7:6$ Parity: $-1$ Primitive: yes Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $1$ Generators: (1,3,2,6,4,5), (1,2,3,4,5,6,7)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$

Resolvents shown for degrees $\leq 47$

## Subfields

Prime degree - none

## Low degree siblings

14T4, 21T4, 42T4

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 $1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $3, 3, 1$ $7$ $3$ $(2,3,5)(4,7,6)$ $6, 1$ $7$ $6$ $(2,4,3,7,5,6)$ $3, 3, 1$ $7$ $3$ $(2,5,3)(4,6,7)$ $6, 1$ $7$ $6$ $(2,6,5,7,3,4)$ $2, 2, 2, 1$ $7$ $2$ $(2,7)(3,6)(4,5)$ $7$ $6$ $7$ $(1,2,3,4,5,6,7)$

## Group invariants

 Order: $42=2 \cdot 3 \cdot 7$ Cyclic: no Abelian: no Solvable: yes GAP id: [42, 1]
 Character table: 2 1 1 1 1 1 1 . 3 1 1 1 1 1 1 . 7 1 . . . . . 1 1a 3a 6a 3b 6b 2a 7a 2P 1a 3b 3a 3a 3b 1a 7a 3P 1a 1a 2a 1a 2a 2a 7a 5P 1a 3b 6b 3a 6a 2a 7a 7P 1a 3a 6a 3b 6b 2a 1a X.1 1 1 1 1 1 1 1 X.2 1 1 -1 1 -1 -1 1 X.3 1 A -/A /A -A -1 1 X.4 1 /A -A A -/A -1 1 X.5 1 A /A /A A 1 1 X.6 1 /A A A /A 1 1 X.7 6 . . . . . -1 A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3