↑

Properties

Label 15T47
Order \(2520\)
n \(15\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $A_7$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $15$
Transitive number $t$ :  $47$
Group :  $A_7$
CHM label :  $A_{7}(15)$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,9,10,3,14)(2,15,7,12,6)(4,5,11,13,8), (1,2,3)(5,6,7)(8,10,9)(12,14,13)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: None

Low degree siblings

7T6, 15T47, 21T33, 35T28, 42T294, 42T299

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy Classes

Cycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $105$ $2$ $( 1, 4)( 3,11)( 5, 8)( 6,14)( 9,12)(13,15)$
$ 4, 4, 4, 2, 1 $ $630$ $4$ $( 1,14, 4, 6)( 3, 9,11,12)( 5,13, 8,15)( 7,10)$
$ 5, 5, 5 $ $504$ $5$ $( 1, 5,14, 7, 2)( 3, 9, 8,15,13)( 4, 6,10,12,11)$
$ 7, 7, 1 $ $360$ $7$ $( 1, 7, 4, 8,13, 2,11)( 3,10,15, 6, 5, 9,12)$
$ 7, 7, 1 $ $360$ $7$ $( 1,11, 2,13, 8, 4, 7)( 3,12, 9, 5, 6,15,10)$
$ 3, 3, 3, 3, 3 $ $70$ $3$ $( 1, 9, 7)( 2, 6,11)( 3,12,15)( 4,14, 5)( 8,13,10)$
$ 6, 6, 3 $ $210$ $6$ $( 1,14,13, 9, 3, 8)( 2,10, 7)( 4, 6,15,12,11, 5)$
$ 3, 3, 3, 3, 1, 1, 1 $ $280$ $3$ $( 1, 3, 4)( 5, 7,15)( 8,10,13)( 9,12,14)$

Group invariants

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

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

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