Properties

Label 35T44
Degree $35$
Order $40320$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $S_8$

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Show commands: Magma

magma: G := TransitiveGroup(35, 44);
 

Group invariants

Abstract group:  $S_8$
magma: IdentifyGroup(G);
 
Order:  $40320=2^{7} \cdot 3^{2} \cdot 5 \cdot 7$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $35$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $44$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
magma: IsEven(G);
 
Primitive:  yes
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,2,4,8,3,6,12)(5,10,15,19,25,29,22)(7,14,18,24,28,34,9)(11,16,20,27,23,13,17)(21,26,32,35,30,31,33)$, $(1,3)(5,6)(9,12)(11,14)(17,23)(18,21)(19,24)(25,31)(27,29)(32,35)$
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

8T50, 16T1838, 28T502, 30T1153

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 $2^{10},1^{15}$ $28$ $2$ $10$ $( 1,28)( 4, 5)( 7,25)( 8,27)(10,11)(12,13)(15,32)(16,24)(17,20)(18,26)$
2B $2^{12},1^{11}$ $105$ $2$ $12$ $( 1,16)( 3,30)( 4,11)( 5,10)( 6,33)( 8,13)(12,27)(14,34)(19,22)(21,35)(23,31)(24,28)$
2C $2^{14},1^{7}$ $210$ $2$ $14$ $( 1,28)( 3,22)( 4, 5)( 6,34)( 7,20)( 8,13)(10,11)(12,27)(14,33)(15,26)(16,24)(17,25)(18,32)(19,30)$
2D $2^{16},1^{3}$ $420$ $2$ $16$ $( 1,16)( 3,30)( 4, 5)( 6,34)( 7,15)( 8,13)(10,11)(12,27)(14,33)(17,18)(19,22)(20,26)(21,31)(23,35)(24,28)(25,32)$
3A $3^{10},1^{5}$ $112$ $3$ $20$ $( 1,28,22)( 4,34, 5)( 7,25,23)( 8,27, 9)(10,33,11)(12,29,13)(15,32,21)(16,30,24)(17,31,20)(18,35,26)$
3B $3^{11},1^{2}$ $1120$ $3$ $22$ $( 1,27,30)( 2, 3,19)( 4,23,26)( 5,25,35)( 7,18,34)( 8,16,22)( 9,24,28)(10,21,20)(11,32,31)(12,29,13)(15,17,33)$
4A $4^{7},2^{3},1$ $420$ $4$ $24$ $( 1,28,22, 3)( 4,34, 6, 5)( 7,31,20,23)( 8,29,13, 9)(10,33,14,11)(12,27)(15,35,26,21)(16,30,19,24)(17,25)(18,32)$
4B $4^{6},2^{4},1^{3}$ $1260$ $4$ $22$ $( 1,16,33, 6)( 3,24,10,34)( 4,22,19,11)( 5,28,30,14)( 7,29)( 8,23)( 9,20)(12,17,27,25)(13,31)(15,21,26,35)$
4C $4^{7},2,1^{5}$ $1260$ $4$ $22$ $( 1,16,22,19)( 3,24,28,30)( 4,33, 6,11)( 5,10,34,14)( 7,23,20,31)( 8,29,13, 9)(12,27)(15,21,26,35)$
4D $4^{7},2^{3},1$ $2520$ $4$ $24$ $( 1,16,22,19)( 3,24,28,30)( 4,34, 6, 5)( 7,21,20,35)( 8,29,13, 9)(10,33,14,11)(12,27)(15,23,26,31)(17,18)(25,32)$
5A $5^{7}$ $1344$ $5$ $28$ $( 1,28,22, 3, 2)( 4,17,25,34, 7)( 5,31,20,23, 6)( 8,16,12,27,30)( 9,19,24,29,13)(10,18,32,33,15)(11,35,26,21,14)$
6A $6^{3},3^{4},2,1^{3}$ $1120$ $6$ $24$ $( 1,28,22)( 2, 3)( 4,23, 5,25,34, 7)( 8,16, 9,24,27,30)(10,21,11,32,33,15)(12,29,13)(17,31,20)(18,35,26)$
6B $6^{3},3^{5},2$ $1120$ $6$ $26$ $( 1,27,30)( 2, 3,19)( 4,21,26,10,23,20)( 5,32,35,11,25,31)( 6,14)( 7,17,34,15,18,33)( 8,16,22)( 9,24,28)(12,29,13)$
6C $6^{4},3^{2},2^{2},1$ $1680$ $6$ $26$ $( 1,28,22)( 2, 3)( 4,23, 5,25,34, 7)( 8,10, 9,11,27,33)(12,35,13,18,29,26)(14,19)(15,16,21,24,32,30)(17,31,20)$
6D $6^{4},3^{3},1^{2}$ $3360$ $6$ $26$ $( 1,29,13, 9,19, 2)( 3,24,28,12,27,30)( 4,18,33, 7,23,20)( 5,35,14)( 6,11,31)( 8,16,22)(10,17,34,15,21,26)$
6E $6^{5},3,2$ $3360$ $6$ $28$ $( 1,29,13, 9,19, 2)( 3,24,28,12,27,30)( 4,17,33,15,23,26)( 5,31,14,11,35, 6)( 7,21,20,10,18,34)( 8,16,22)(25,32)$
7A $7^{5}$ $5760$ $7$ $30$ $( 1,29, 6, 5,31,14, 2)( 3,24,35,13, 4,17,33)( 7,21,19,11,28,12,34)( 8,23,26, 9,20,10,22)(15,16,18,27,25,32,30)$
8A $8^{3},4^{2},2,1$ $5040$ $8$ $28$ $( 1,29, 6, 2)( 3,24,35,14, 5,28,12,34)( 4,22, 8,23,19,11,31,13)( 7,16,18,33)( 9,20)(10,17,27,25,30,15,21,26)$
10A $10^{2},5^{3}$ $4032$ $10$ $30$ $( 1,28,22, 3, 2)( 4,17,25,34, 7)( 5,31,20,23, 6)( 8,10,12,32,30,15,16,18,27,33)( 9,14,24,35,13,21,19,11,29,26)$
12A $12^{2},6,4,1$ $3360$ $12$ $30$ $( 1,16,33, 3,24,10,22,19,11,28,30,14)( 4,34, 6, 5)( 7,21, 8,23,26, 9,20,35,13,31,15,29)(12,17,18,27,25,32)$
15A $15^{2},5$ $2688$ $15$ $32$ $( 1,28,22, 3, 2)( 4,12,32,34, 8,10,17,27,33, 7,16,18,25,30,15)( 5,29,26,23,19,11,31,13,21, 6,24,35,20, 9,14)$

Malle's constant $a(G)$:     $1/10$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 5A 6A 6B 6C 6D 6E 7A 8A 10A 12A 15A
Size 1 28 105 210 420 112 1120 420 1260 1260 2520 1344 1120 1120 1680 3360 3360 5760 5040 4032 3360 2688
2 P 1A 1A 1A 1A 1A 3A 3B 2C 2B 2C 2C 5A 3A 3B 3A 3B 3B 7A 4B 5A 6C 15A
3 P 1A 2A 2B 2C 2D 1A 1A 4A 4B 4C 4D 5A 2A 2A 2C 2B 2D 7A 8A 10A 4A 5A
5 P 1A 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 1A 6A 6B 6C 6D 6E 7A 8A 2A 12A 3A
7 P 1A 2A 2B 2C 2D 3A 3B 4A 4B 4C 4D 5A 6A 6B 6C 6D 6E 1A 8A 10A 12A 15A
Type

magma: CharacterTable(G);
 

Regular extensions

Data not computed