Properties

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

Related objects

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Group action invariants

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

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  1  .  .  .  .  2  1  2
     5  1  .  .  .  .  1  .  .  .
     7  1  .  .  1  1  .  .  .  .

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

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