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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

$\card{(G/N)}$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 TypeSizeOrderIndexRepresentative
1A $1^{35}$ $1$ $1$ $0$ $()$
2A $2^{10},1^{15}$ $21$ $2$ $10$ $( 1, 5)( 3, 6)( 4,28)( 8,26)( 9,35)(12,32)(13,15)(18,27)(21,29)(22,34)$
2B $2^{14},1^{7}$ $105$ $2$ $14$ $( 1, 5)( 2,11)( 3,13)( 4,35)( 6,15)( 8,26)( 9,28)(12,32)(16,23)(18,34)(19,20)(21,29)(22,27)(25,30)$
2C $2^{16},1^{3}$ $105$ $2$ $16$ $( 1,32)( 2,25)( 3,28)( 4, 6)( 5,12)( 7,10)( 8,26)( 9,13)(11,30)(14,31)(15,35)(16,23)(18,34)(19,20)(21,29)(22,27)$
3A $3^{10},1^{5}$ $70$ $3$ $20$ $( 1, 5, 2)( 3, 7, 6)( 4,28,31)( 8,26,19)( 9,10,35)(12,32,30)(13,14,15)(16,29,21)(17,34,22)(18,27,33)$
3B $3^{11},1^{2}$ $280$ $3$ $22$ $( 1,27,30)( 2,18,32)( 3,16,35)( 4,26,14)( 5,33,12)( 6,21,10)( 7,29, 9)( 8,13,31)(11,25,24)(15,28,19)(17,34,22)$
4A $4^{7},2^{3},1$ $210$ $4$ $24$ $( 1, 5, 2,11)( 3,13)( 4,10,35,31)( 6,14,15, 7)( 8,26,19,20)( 9,28)(12,32,30,25)(16,23,29,21)(17,34,33,18)(22,27)$
4B $4^{7},2^{3},1$ $630$ $4$ $24$ $( 1,32, 2,25)( 3,28)( 4, 7,35,14)( 5,30,11,12)( 6,31,15,10)( 8,26,19,20)( 9,13)(16,23,29,21)(17,34,33,18)(22,27)$
5A $5^{7}$ $504$ $5$ $28$ $( 1, 5, 2,11,24)( 3,13,19,15,26)( 4,10,23,29,31)( 6,14,20, 8, 7)( 9,16,35,21,28)(12,17,34,33,25)(18,32,22,27,30)$
6A $6^{4},3^{2},2^{2},1$ $210$ $6$ $26$ $( 1, 5, 2)( 3, 7, 6)( 4,22,31,34,28,17)( 8,13,19,15,26,14)( 9,30,35,32,10,12)(11,24)(16,18,21,33,29,27)(23,25)$
6B $6^{3},3^{4},2,1^{3}$ $420$ $6$ $24$ $( 1, 5, 2)( 3, 7, 6)( 4,28,31)( 8,13,19,15,26,14)( 9,16,35,21,10,29)(11,24)(12,27,30,18,32,33)(17,34,22)$
6C $6^{5},3,2$ $840$ $6$ $28$ $( 1,17,34,33,25,24)( 2,18,32)( 3,16,35,14, 4,26)( 5,22,27,30,11,12)( 6,21,28,19,15,10)( 7,23,29,31,20, 8)( 9,13)$
7A $7^{5}$ $720$ $7$ $30$ $( 1,17,20, 8, 7,23,24)( 2,18,14, 4,33, 6,21)( 3,16,11,12,31,34,26)( 5,22,19,15,10,25,29)( 9,30,35,32,28,27,13)$
10A $10^{2},5^{3}$ $504$ $10$ $30$ $( 1, 5, 2,11,24)( 3,13,19,15,26)( 4,33,23,12,31,34,10,25,29,17)( 6,14,20, 8, 7)( 9,30,35,32,28,27,16,18,21,22)$
12A $12^{2},6,4,1$ $420$ $12$ $30$ $( 1,32,16,11,12,21, 2,25,29, 5,30,23)( 3,28,27,13, 9,22)( 4,33, 6,31,34, 7,35,17,15,10,18,14)( 8,26,19,20)$

Malle's constant $a(G)$:     $1/10$

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:

1A 2A 2B 2C 3A 3B 4A 4B 5A 6A 6B 6C 7A 10A 12A
Size 1 21 105 105 70 280 210 630 504 210 420 840 720 504 420
2 P 1A 1A 1A 1A 3A 3B 2B 2B 5A 3A 3A 3B 7A 5A 6A
3 P 1A 2A 2B 2C 1A 1A 4A 4B 5A 2B 2A 2C 7A 10A 4A
5 P 1A 2A 2B 2C 3A 3B 4A 4B 1A 6A 6B 6C 7A 2A 12A
7 P 1A 2A 2B 2C 3A 3B 4A 4B 5A 6A 6B 6C 1A 10A 12A
Type

magma: CharacterTable(G);