Properties

Label 7T4
Order \(42\)
n \(7\)
Cyclic No
Abelian No
Solvable Yes
Primitive Yes
$p$-group No
Group: $F_7$

Related objects

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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)
Generators:  (1,3,2,6,4,5), (1,2,3,4,5,6,7)
$|\Aut(F/K)|$:  $1$

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 TypeSizeOrderRepresentative
$ 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