Properties

Label 7T6
Degree $7$
Order $2520$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $A_7$

Related objects

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Group action invariants

Degree $n$:  $7$
Transitive number $t$:  $6$
Group:  $A_7$
CHM label:  $A7$
Parity:  $1$
Primitive:  yes
Nilpotency class:  $-1$ (not nilpotent)
$|\Aut(F/K)|$:  $1$
Generators:  (3,4,5,6,7), (1,2,3)

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Prime degree - none

Low degree siblings

15T47 x 2, 21T33, 35T28, 42T294, 42T299

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 $ $1$ $1$ $()$
$ 2, 2, 1, 1, 1 $ $105$ $2$ $(1,4)(2,7)$
$ 4, 2, 1 $ $630$ $4$ $(1,2,4,7)(3,5)$
$ 5, 1, 1 $ $504$ $5$ $(2,7,6,4,3)$
$ 7 $ $360$ $7$ $(1,3,7,2,6,4,5)$
$ 7 $ $360$ $7$ $(1,5,4,6,2,7,3)$
$ 3, 3, 1 $ $280$ $3$ $(2,7,5)(3,6,4)$
$ 3, 1, 1, 1, 1 $ $70$ $3$ $(3,4,6)$
$ 3, 2, 2 $ $210$ $6$ $(1,5)(2,7)(3,4,6)$

Group invariants

Order:  $2520=2^{3} \cdot 3^{2} \cdot 5 \cdot 7$
Cyclic:  no
Abelian:  no
Solvable:  no
GAP id:  not available
Character table:   
     2  3  .  .  2  3  2  2  .  .
     3  2  .  .  2  1  1  .  2  .
     5  1  .  .  .  .  .  .  .  1
     7  1  1  1  .  .  .  .  .  .

       1a 7a 7b 3a 2a 6a 4a 3b 5a
    2P 1a 7a 7b 3a 1a 3a 2a 3b 5a
    3P 1a 7b 7a 1a 2a 2a 4a 1a 5a
    5P 1a 7b 7a 3a 2a 6a 4a 3b 1a
    7P 1a 1a 1a 3a 2a 6a 4a 3b 5a

X.1     1  1  1  1  1  1  1  1  1
X.2     6 -1 -1  3  2 -1  .  .  1
X.3    10  A /A  1 -2  1  .  1  .
X.4    10 /A  A  1 -2  1  .  1  .
X.5    14  .  .  2  2  2  . -1 -1
X.6    14  .  . -1  2 -1  .  2 -1
X.7    15  1  1  3 -1 -1 -1  .  .
X.8    21  .  . -3  1  1 -1  .  1
X.9    35  .  . -1 -1 -1  1 -1  .

A = E(7)^3+E(7)^5+E(7)^6
  = (-1-Sqrt(-7))/2 = -1-b7