↑

Properties

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

Learn more about

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

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

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  .  .  .  2 -1
X.7    15 -1  3 -1 -1  1  1  .  .
X.8    21  1 -3  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