Properties

Label 35T28
Order \(2520\)
n \(35\)
Cyclic No
Abelian No
Solvable No
Primitive Yes
$p$-group No
Group: $A_7$

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Group action invariants

Degree $n$ :  $35$
Transitive number $t$ :  $28$
Group :  $A_7$
Parity:  $1$
Primitive:  Yes
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,2,4,8,3,6,12)(5,10,15,19,25,29,22)(7,14,18,24,28,34,9)(11,16,20,27,23,13,17)(21,26,32,35,30,31,33), (2,5,11)(3,7,15)(4,9,10)(6,13,14)(16,21,23)(17,18,22)(19,26,20)(25,30,32)(27,33,34)(28,31,35)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

None

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Degree 7: None

Low degree siblings

7T6, 15T47 x 2, 21T33, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 5, 5, 5, 5, 5, 5, 5 $ $504$ $5$ $( 1,22,20, 4, 5)( 2,17,28,34,31)( 3,23,25, 6, 7)( 8, 9,11,18,26) (10,27,35,33,19)(12,13,15,21,24)(14,16,32,29,30)$
$ 7, 7, 7, 7, 7 $ $360$ $7$ $( 1, 2,34,15, 3, 6,35)( 4,32, 7,19,29,11,22)( 5,30, 8,14,23,18,28) ( 9,12,10,17,16,27,13)(20,21,25,26,31,24,33)$
$ 7, 7, 7, 7, 7 $ $360$ $7$ $( 1,35, 6, 3,15,34, 2)( 4,22,11,29,19, 7,32)( 5,28,18,23,14, 8,30) ( 9,13,27,16,17,10,12)(20,33,24,31,26,25,21)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $280$ $3$ $( 1,13,34)( 2,12,20)( 3, 9,23)( 4,24,14)( 5,29,15)( 6, 8,11)( 7,27,35) (10,25,19)(16,32,31)(17,28,30)(18,26,33)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ $70$ $3$ $( 1,17,12)( 2,34,30)( 3,23, 9)( 4,15,16)( 5,32,24)( 6,35,19)( 7,10, 8) (11,27,25)(13,28,20)(14,29,31)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $ $105$ $2$ $( 1,12)( 2,30)( 3, 9)( 4, 7)( 5,25)( 6,31)(10,15)(11,32)(13,28)(14,35)(17,18) (20,26)(21,23)(33,34)$
$ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ $630$ $4$ $( 1,26,12,20)( 2,33,30,34)( 3,21, 9,23)( 4,11, 7,32)( 5,10,25,15)( 6,14,31,35) ( 8,24)(13,17,28,18)(16,27)(19,29)$
$ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ $210$ $6$ $( 1,26, 2,18,28,33)( 3,21)( 4,25,31, 7, 5, 6)( 8,24,19,16,27,29)( 9,23) (10,11,35)(12,20,30,17,13,34)(14,15,32)$

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

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