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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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Show commands: Magma

magma: G := TransitiveGroup(35, 28);
 

Group action invariants

Degree $n$:  $35$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $28$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $A_7$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
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)
magma: Generators(G);
 

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $2520=2^{3} \cdot 3^{2} \cdot 5 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  2520.a
magma: IdentifyGroup(G);
 
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 3b 5a 7a 7b
    2P 1a 1a 2a 3a 3a 3b 5a 7a 7b
    3P 1a 2a 4a 1a 2a 1a 5a 7b 7a
    5P 1a 2a 4a 3a 6a 3b 1a 7b 7a
    7P 1a 2a 4a 3a 6a 3b 5a 1a 1a

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

magma: CharacterTable(G);