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Properties

Label 35T36
Degree $35$
Order $20160$
Cyclic no
Abelian no
Solvable no
Primitive yes
$p$-group no
Group: $A_8$

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Show commands: Magma

magma: G := TransitiveGroup(35, 36);
 

Group invariants

Abstract group:  $A_8$
magma: IdentifyGroup(G);
 
Order:  $20160=2^{6} \cdot 3^{2} \cdot 5 \cdot 7$
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 $t$:  $36$
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:  $(1,2,4,8,3,6,12)(5,10,15,19,25,29,22)(7,14,18,24,28,34,9)(11,16,20,27,23,13,17)(21,26,32,35,30,31,33)$, $(1,3,7)(2,6,5)(9,10,12)(11,14,13)(16,18,21)(17,22,23)(19,26,24)(25,31,28)(27,33,29)(30,35,32)$
magma: Generators(G);
 

Low degree resolvents

none

Resolvents shown for degrees $\leq 47$

Subfields

Degree 5: None

Degree 7: None

Low degree siblings

8T49, 15T72 x 2, 28T433

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 $2^{12},1^{11}$ $105$ $2$ $12$ $( 1, 9)( 2,13)( 3,12)( 4, 7)(10,15)(17,21)(18,23)(19,29)(20,33)(24,27)(26,34)(28,30)$
2B $2^{14},1^{7}$ $210$ $2$ $14$ $( 1,12)( 2,30)( 3, 9)( 4, 7)( 5,25)( 6,31)(10,15)(11,32)(13,28)(14,35)(17,18)(20,26)(21,23)(33,34)$
3A $3^{10},1^{5}$ $112$ $3$ $20$ $( 1,18,12)( 2,33,30)( 3,21, 9)( 4, 8,15)( 5,27,32)( 6,29,35)( 7,16,10)(11,25,24)(13,28,26)(14,31,19)$
3B $3^{11},1^{2}$ $1120$ $3$ $22$ $( 1,16, 9)( 2,27,35)( 3,18,10)( 4, 8,15)( 5,29,30)( 6,33,32)( 7,21,12)(11,28,19)(13,31,24)(14,25,26)(17,23,22)$
4A $4^{6},2^{4},1^{3}$ $1260$ $4$ $22$ $( 1,21,34,13)( 2,26,17, 9)( 3,33,20,12)( 4, 6, 7,31)( 8,14)(10,19)(11,24,32,27)(15,29)(16,35)(18,23,30,28)$
4B $4^{7},2^{3},1$ $2520$ $4$ $24$ $( 1,21,17, 9)( 2,33,34,30)( 3,18,23,12)( 4, 7)( 5,31)( 6,25)( 8,15,16,10)(11,29,32,19)(13,28,26,20)(14,27,35,24)$
5A $5^{7}$ $1344$ $5$ $28$ $( 1,18,17,22,12)( 2,25, 5,33,32)( 3, 4, 7,21,10)( 6,31,26,14,28)( 8,15,16,23, 9)(11,27,34,30,24)(13,29,35,19,20)$
6A $6^{4},3^{2},2^{2},1$ $1680$ $6$ $26$ $( 1,18,12)( 2,16,30, 7,33,10)( 3,27, 9, 5,21,32)( 4,24,15,25, 8,11)( 6,29,35)(13,31,26,14,28,19)(17,22)(23,34)$
6B $6^{4},3^{3},1^{2}$ $3360$ $6$ $26$ $( 1, 4, 7,21,17, 9)( 2, 6,26,32,27,11)( 3,18,10)( 8,15,16,23,22,12)(13,24,34)(14,19,35,28,25,33)(20,30,29)$
7A1 $7^{5}$ $2880$ $7$ $30$ $( 1, 4,26,14,27,23,30)( 2,18,10,19,11,16,35)( 3,25, 7,31, 5,21,32)( 6,33,17, 9,24,15,28)( 8,20,13,29,34,22,12)$
7A-1 $7^{5}$ $2880$ $7$ $30$ $( 1, 8,20,28,29,34,22)( 2,12,15,26,31,27,23)( 3,24, 4,13,35,33,17)( 5, 9,11,10,14,32,21)( 6,30,18, 7,19,25,16)$
15A1 $15^{2},5$ $1344$ $15$ $32$ $( 1,18,17,22,12)( 2, 6, 7,33,14, 3,25,31,21,32,28, 4, 5,26,10)( 8,11,19,23,30,29,15,27,20, 9,24,35,16,34,13)$
15A-1 $15^{2},5$ $1344$ $15$ $32$ $( 1,12,18,17,22)( 2,29, 4,27,20, 3,24, 6,16,34,28, 8,25,19,23)( 5,13,15,32,26, 7,30,35,10,33,31, 9,11,14,21)$

Malle's constant $a(G)$:     $1/12$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 2B 3A 3B 4A 4B 5A 6A 6B 7A1 7A-1 15A1 15A-1
Size 1 105 210 112 1120 1260 2520 1344 1680 3360 2880 2880 1344 1344
2 P 1A 1A 1A 3A 3B 2A 2B 5A 3A 3B 7A1 7A-1 15A1 15A-1
3 P 1A 2A 2B 1A 1A 4A 4B 5A 2B 2A 7A-1 7A1 5A 5A
5 P 1A 2A 2B 3A 3B 4A 4B 1A 6A 6B 7A-1 7A1 3A 3A
7 P 1A 2A 2B 3A 3B 4A 4B 5A 6A 6B 1A 1A 15A-1 15A1
Type

magma: CharacterTable(G);
 

Regular extensions

Data not computed