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

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  2  1
     5  1  .  .  1  .  .  .  .  .
     7  1  1  1  .  .  .  .  .  .

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

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  . -1  2  2
X.6    14  .  . -1  2  .  2 -1 -1
X.7    15  1  1  . -1 -1  .  3 -1
X.8    21  .  .  1  1 -1  . -3  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);