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Properties

Label 35T31
Degree $35$
Order $5040$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $S_7$

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Show commands: Magma

magma: G := TransitiveGroup(35, 31);
 

Group action invariants

Degree $n$:  $35$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $31$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_7$
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)(3,7)(9,10)(13,14)(16,21)(17,22)(19,26)(27,33)(28,31)(30,32)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Degree 7: None

Low degree siblings

7T7, 14T46, 21T38, 30T565, 42T411, 42T412, 42T413, 42T418

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle 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 $ $105$ $2$ $( 2,28)( 4,16)( 5,29)( 6,27)( 7, 8)( 9,23)(11,35)(12,17)(13,34)(14,32)(19,25) (20,30)(24,31)(26,33)$
$ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ $630$ $4$ $( 1,21)( 2, 9,28,23)( 3,22)( 4,35,16,11)( 5,31,29,24)( 6,30,27,20) ( 7,33, 8,26)(12,32,17,14)(13,25,34,19)(15,18)$
$ 5, 5, 5, 5, 5, 5, 5 $ $504$ $5$ $( 1,11,27,34,22)( 2,18, 5,33,17)( 3,15,13,29,23)( 4,26,31,10,19) ( 6,21, 7,14,16)( 8,20,28,35, 9)(12,24,25,32,30)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $21$ $2$ $( 4,16)( 5,24)( 6,19)( 7, 8)( 9,23)(12,17)(13,20)(25,27)(29,31)(30,34)$
$ 10, 10, 5, 5, 5 $ $504$ $10$ $( 1,18,33,10,28)( 2,35,22,11,26)( 3,15,21,14,32)( 4,29,17,25, 8,16,31,12,27, 7 )( 5, 6,34,23,20,24,19,30, 9,13)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $ $70$ $3$ $( 1,17,22)( 2, 5,34)( 3, 7,23)( 8,10, 9)(11,27,33)(13,29,14)(15,16,21) (19,26,35)(20,28,31)(24,32,30)$
$ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $ $210$ $4$ $( 1,23)( 2,16,28,10)( 3,17)( 5,21,31, 9)( 6,18,25,12)( 7,22)( 8,34,15,20) (11,24,29,35)(13,26,33,30)(14,19,27,32)$
$ 6, 6, 6, 6, 3, 3, 2, 2, 1 $ $210$ $6$ $( 1,22,17)( 2,20, 5,28,34,31)( 3,23, 7)( 6,25)( 8,21,10,15, 9,16) (11,13,27,29,33,14)(12,18)(19,24,26,32,35,30)$
$ 12, 12, 6, 4, 1 $ $420$ $12$ $( 1, 3,22,23,17, 7)( 2, 9,20,16, 5, 8,28,21,34,10,31,15)( 6,12,25,18) (11,19,13,24,27,26,29,32,33,35,14,30)$
$ 6, 6, 6, 3, 3, 3, 3, 2, 1, 1, 1 $ $420$ $6$ $( 1,22,17)( 2,34, 5)( 3,23, 7)( 8,21,10,15, 9,16)(11,30,27,24,33,32)(12,18) (13,19,29,26,14,35)(20,31,28)$
$ 7, 7, 7, 7, 7 $ $720$ $7$ $( 1,28,13, 7, 3,27, 5)( 2,29,26,15,25, 9,17)( 4,12,10,14,33,21,18) ( 6, 8,32,11,31,22,16)(19,34,24,35,20,30,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $105$ $2$ $( 1,24)( 2,11)( 3,20)( 4,27)( 6,13)( 7,26)( 9,34)(10,33)(12,29)(15,19)(16,18) (17,21)(22,23)(25,28)(30,35)(31,32)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1 $ $280$ $3$ $( 1,10, 6)( 2, 7,12)( 3,17, 4)( 5, 8,14)( 9,19,31)(11,26,29)(13,24,33) (15,32,34)(16,25,22)(18,28,23)(20,21,27)$
$ 6, 6, 6, 6, 6, 3, 2 $ $840$ $6$ $( 1,13,10,24, 6,33)( 2,29, 7,11,12,26)( 3,27,17,20, 4,21)( 5,14, 8) ( 9,32,19,34,31,15)(16,23,25,18,22,28)(30,35)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $5040=2^{4} \cdot 3^{2} \cdot 5 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  5040.w
magma: IdentifyGroup(G);
 
Character table:

Size
2 P
3 P
5 P
7 P
Type

magma: CharacterTable(G);