Properties

Label 35T28
Degree $35$
Order $2520$
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)
$|\Aut(F/K)|$:  $1$
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)

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

Group invariants

Order:  $2520=2^{3} \cdot 3^{2} \cdot 5 \cdot 7$
Cyclic:  no
Abelian:  no
Solvable:  no
GAP id:  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 3b 4a
    2P 1a 5a 7a 7b 1a 3a 3a 3b 2a
    3P 1a 5a 7b 7a 2a 1a 2a 1a 4a
    5P 1a 1a 7b 7a 2a 3a 6a 3b 4a
    7P 1a 5a 1a 1a 2a 3a 6a 3b 4a

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