↑

Properties

Label 10T35
10T35 1 2 1->2 1->2 7 1->7 5 2->5 10 2->10 3 4 3->4 3->4 6 3->6 4->2 4->5 4->7 4->7 5->3 5->6 8 5->8 5->8 6->7 6->8 7->3 7->8 8->1 8->6 9 9->10 10->1
Degree $10$
Order $1440$
Cyclic no
Abelian no
Solvable no
Transitivity $3$
Primitive yes
$p$-group no
Group: $(A_6 : C_2) : C_2$

Related objects

Downloads

Learn more

Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(10, 35);
 
Copy content sage:G = TransitiveGroup(10, 35)
 
Copy content oscar:G = transitive_group(10, 35)
 
Copy content gap:G := TransitiveGroup(10, 35);
 

Group invariants

Abstract group:  $(A_6 : C_2) : C_2$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Copy content oscar:small_group_identification(G)
 
Copy content gap:IdGroup(G);
 
Order:  $1440=2^{5} \cdot 3^{2} \cdot 5$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar:order(G)
 
Copy content gap:Order(G);
 
Cyclic:  no
Copy content comment:Determine if group is cyclic
 
Copy content magma:IsCyclic(G);
 
Copy content sage:G.is_cyclic()
 
Copy content oscar:is_cyclic(G)
 
Copy content gap:IsCyclic(G);
 
Abelian:  no
Copy content comment:Determine if group is abelian
 
Copy content magma:IsAbelian(G);
 
Copy content sage:G.is_abelian()
 
Copy content oscar:is_abelian(G)
 
Copy content gap:IsAbelian(G);
 
Solvable:  no
Copy content comment:Determine if group is solvable
 
Copy content magma:IsSolvable(G);
 
Copy content sage:G.is_solvable()
 
Copy content oscar:is_solvable(G)
 
Copy content gap:IsSolvable(G);
 
Nilpotency class:   not nilpotent
Copy content comment:Nilpotency class
 
Copy content magma:NilpotencyClass(G);
 
Copy content sage:libgap(G).NilpotencyClassOfGroup() if G.is_nilpotent() else -1
 
Copy content oscar:if is_nilpotent(G) nilpotency_class(G) end
 
Copy content gap:if IsNilpotentGroup(G) then NilpotencyClassOfGroup(G); fi;
 

Group action invariants

Degree $n$:  $10$
Copy content comment:Degree
 
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Copy content sage:G.degree()
 
Copy content oscar:degree(G)
 
Copy content gap:NrMovedPoints(G);
 
Transitive number $t$:  $35$
Copy content comment:Transitive number
 
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Copy content sage:G.transitive_number()
 
Copy content oscar:transitive_group_identification(G)[2]
 
Copy content gap:TransitiveIdentification(G);
 
CHM label:   $L(10).2^{2}=P|L(2,9)$
Parity:  $-1$
Copy content comment:Parity
 
Copy content magma:IsEven(G);
 
Copy content sage:all(g.SignPerm() == 1 for g in libgap(G).GeneratorsOfGroup())
 
Copy content oscar:is_even(G)
 
Copy content gap:ForAll(GeneratorsOfGroup(G), g -> SignPerm(g) = 1);
 
Transitivity:  3
Primitive:  yes
Copy content comment:Determine if group is primitive
 
Copy content magma:IsPrimitive(G);
 
Copy content sage:G.is_primitive()
 
Copy content oscar:is_primitive(G)
 
Copy content gap:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(10).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(10), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(10), G));
 
Generators:  $(1,2)(4,7)(5,8)(9,10)$, $(1,2,10)(3,4,5)(6,7,8)$, $(1,7,3,4,2,5,6,8)$, $(3,6)(4,7)(5,8)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 
Copy content gap:GeneratorsOfGroup(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: None

Low degree siblings

12T220, 20T201, 20T204, 20T208, 24T2960, 30T264, 36T2341, 40T1198, 40T1199, 40T1201, 45T187

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{10}$ $1$ $1$ $0$ $()$
2A $2^{3},1^{4}$ $30$ $2$ $3$ $(1,6)(2,3)(4,8)$
2B $2^{5}$ $36$ $2$ $5$ $( 1, 5)( 2, 7)( 3, 4)( 6, 8)( 9,10)$
2C $2^{4},1^{2}$ $45$ $2$ $4$ $(1,6)(2,4)(3,8)(5,9)$
3A $3^{3},1$ $80$ $3$ $6$ $( 1, 2, 4)( 3, 8, 6)( 7, 9,10)$
4A $4^{2},2$ $90$ $4$ $7$ $( 1, 9, 6, 5)( 2, 8, 4, 3)( 7,10)$
4B $4^{2},1^{2}$ $90$ $4$ $6$ $(1,2,7,8)(3,5,9,4)$
4C $4^{2},1^{2}$ $180$ $4$ $6$ $(1,7,3,2)(5,9,6,8)$
5A $5^{2}$ $144$ $5$ $8$ $( 1, 2, 8,10, 4)( 3, 5, 7, 6, 9)$
6A $6,3,1$ $240$ $6$ $7$ $( 1, 8, 2, 6, 4, 3)( 7,10, 9)$
8A $8,2$ $180$ $8$ $8$ $( 1, 9, 2, 4, 7, 3, 8, 5)( 6,10)$
8B $8,1^{2}$ $180$ $8$ $7$ $( 1, 7, 9, 3,10, 4, 2, 6)$
10A $10$ $144$ $10$ $9$ $( 1, 9, 2, 3, 8, 5,10, 7, 4, 6)$

Malle's constant $a(G)$:     $1/3$

Copy content comment:Conjugacy classes
 
Copy content magma:ConjugacyClasses(G);
 
Copy content sage:G.conjugacy_classes()
 
Copy content oscar:conjugacy_classes(G)
 
Copy content gap:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 3A 4A 4B 4C 5A 6A 8A 8B 10A
Size 1 30 36 45 80 90 90 180 144 240 180 180 144
2 P 1A 1A 1A 1A 3A 2C 2C 2C 5A 3A 4B 4B 5A
3 P 1A 2A 2B 2C 1A 4A 4B 4C 5A 2A 8A 8B 10A
5 P 1A 2A 2B 2C 3A 4A 4B 4C 1A 6A 8A 8B 2B
Type
1440.5841.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
1440.5841.1b R 1 −1 −1 1 1 −1 1 1 1 −1 1 −1 −1
1440.5841.1c R 1 −1 1 1 1 −1 1 −1 1 −1 −1 1 1
1440.5841.1d R 1 1 −1 1 1 1 1 −1 1 1 −1 −1 −1
1440.5841.9a R 9 3 −1 1 0 −1 1 1 −1 0 −1 1 −1
1440.5841.9b R 9 3 1 1 0 −1 1 −1 −1 0 1 −1 1
1440.5841.9c R 9 −3 1 1 0 1 1 1 −1 0 −1 −1 1
1440.5841.9d R 9 −3 −1 1 0 1 1 −1 −1 0 1 1 −1
1440.5841.10a R 10 2 0 2 1 2 −2 0 0 −1 0 0 0
1440.5841.10b R 10 −2 0 2 1 −2 −2 0 0 1 0 0 0
1440.5841.16a R 16 0 4 0 −2 0 0 0 1 0 0 0 −1
1440.5841.16b R 16 0 −4 0 −2 0 0 0 1 0 0 0 1
1440.5841.20a R 20 0 0 −4 2 0 0 0 0 0 0 0 0

Copy content comment:Character table
 
Copy content magma:CharacterTable(G);
 
Copy content sage:G.character_table()
 
Copy content oscar:character_table(G)
 
Copy content gap:CharacterTable(G);
 

Regular extensions

$f_{ 1 } =$ $x^{10} + 2 x^{9} + 9 x^{8} + t x^{2} + 2 t x + t$ Copy content Toggle raw display
$f_{ 2 } =$ $\left(5 x^{2}-81\right)^{4} \left(5 x^{2}+50 x+189\right)-2^{14} 3^{12} t$ Copy content Toggle raw display
$f_{ 3 } =$ $x^{8} \left(x-3\right)^{2}-27 t \left(3 x^{2}-2 x+3\right)$ Copy content Toggle raw display
$f_{ 4 } =$ $16 \left(1-t\right) x^{2} \left(x^{2}+5 x+5\right)^{4}+t \left(4 x^{5}+40 x^{4}+140 x^{3}+200 x^{2}+105 x+34\right)^{2}+\left(t-1\right) t \left(5 x+2\right)^{2}$ Copy content Toggle raw display