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

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

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