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Properties

Label 10T38
10T38 1 3 1->3 2 4 2->4 2->4 7 2->7 5 3->5 6 4->6 5->7 10 5->10 5->10 8 6->8 9 7->9 7->9 8->10 9->1 10->2
Degree $10$
Order $1920$
Cyclic no
Abelian no
Solvable no
Transitivity $1$
Primitive no
$p$-group no
Group: $(C_2^4:A_5) : C_2$

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Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(10, 38);
 
Copy content sage:G = TransitiveGroup(10, 38)
 
Copy content oscar:G = transitive_group(10, 38)
 
Copy content gap:G := TransitiveGroup(10, 38);
 

Group invariants

Abstract group:  $(C_2^4:A_5) : C_2$
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:  $1920=2^{7} \cdot 3 \cdot 5$
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 $t$:  $38$
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:   $1/2[2^{5}]S(5)$
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:  1
Primitive:  no
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);
 
$\card{\Aut(F/K)}$:  $2$
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:  $(2,4)(5,10)(7,9)$, $(1,3,5,7,9)(2,4,6,8,10)$, $(2,7)(5,10)$
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)
$2$:  $C_2$
$120$:  $S_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: $S_5$

Low degree siblings

10T37, 16T1328, 20T218, 20T219, 20T222, 20T223, 20T226, 30T329, 30T332, 30T333, 30T341, 32T97736, 40T1581, 40T1582, 40T1583, 40T1584, 40T1587, 40T1588, 40T1595, 40T1596, 40T1658, 40T1659, 40T1676, 40T1677, 40T1678

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 $2^{4},1^{2}$ $5$ $2$ $4$ $(1,6)(2,7)(3,8)(4,9)$
2B $2^{2},1^{6}$ $10$ $2$ $2$ $(2,7)(4,9)$
2C $2^{5}$ $20$ $2$ $5$ $( 1, 3)( 2, 7)( 4, 9)( 5,10)( 6, 8)$
2D $2^{4},1^{2}$ $60$ $2$ $4$ $( 1, 9)( 3,10)( 4, 6)( 5, 8)$
2E $2^{3},1^{4}$ $60$ $2$ $3$ $(1,8)(2,7)(3,6)$
3A $3^{2},1^{4}$ $80$ $3$ $4$ $( 1,10, 8)( 3, 6, 5)$
4A $4,1^{6}$ $20$ $4$ $3$ $(2,9,7,4)$
4B $4,2^{2},1^{2}$ $60$ $4$ $5$ $(1,4,6,9)(2,7)(3,8)$
4C $4^{2},1^{2}$ $60$ $4$ $6$ $( 2, 8, 7, 3)( 4, 5, 9,10)$
4D $4,2^{3}$ $120$ $4$ $6$ $( 1, 4, 6, 9)( 2,10)( 3, 8)( 5, 7)$
4E $4^{2},2$ $240$ $4$ $7$ $( 1, 8, 9, 5)( 2, 7)( 3, 4,10, 6)$
5A $5^{2}$ $384$ $5$ $8$ $( 1, 8,10, 2, 9)( 3, 5, 7, 4, 6)$
6A $3^{2},2^{2}$ $80$ $6$ $6$ $( 1, 8,10)( 2, 7)( 3, 5, 6)( 4, 9)$
6B $6,2^{2}$ $160$ $6$ $7$ $( 1, 3)( 2, 4,10, 7, 9, 5)( 6, 8)$
6C $6,2,1^{2}$ $160$ $6$ $6$ $(1,3,7,6,8,2)(4,9)$
8A $8,1^{2}$ $240$ $8$ $7$ $( 2, 5, 8, 9, 7,10, 3, 4)$
12A $4,3^{2}$ $160$ $12$ $7$ $( 1,10, 8)( 2, 4, 7, 9)( 3, 6, 5)$

Malle's constant $a(G)$:     $1/2$

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 2D 2E 3A 4A 4B 4C 4D 4E 5A 6A 6B 6C 8A 12A
Size 1 5 10 20 60 60 80 20 60 60 120 240 384 80 160 160 240 160
2 P 1A 1A 1A 1A 1A 1A 3A 2B 2B 2A 2B 2D 5A 3A 3A 3A 4C 6A
3 P 1A 2A 2B 2C 2D 2E 1A 4A 4B 4C 4D 4E 5A 2B 2C 2A 8A 4A
5 P 1A 2A 2B 2C 2D 2E 3A 4A 4B 4C 4D 4E 1A 6A 6B 6C 8A 12A
Type
1920.240996.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1920.240996.1b R 1 1 1 −1 1 −1 1 −1 −1 1 1 −1 1 1 −1 1 −1 −1
1920.240996.4a R 4 4 4 2 0 2 1 2 2 0 0 0 −1 1 −1 1 0 −1
1920.240996.4b R 4 4 4 −2 0 −2 1 −2 −2 0 0 0 −1 1 1 1 0 1
1920.240996.5a R 5 5 5 −1 1 −1 −1 −1 −1 1 1 1 0 −1 −1 −1 1 −1
1920.240996.5b R 5 5 5 1 1 1 −1 1 1 1 1 −1 0 −1 1 −1 −1 1
1920.240996.5c R 5 −3 1 3 1 −1 2 −3 1 1 −1 1 0 −2 0 0 −1 0
1920.240996.5d R 5 −3 1 −3 1 1 2 3 −1 1 −1 −1 0 −2 0 0 1 0
1920.240996.6a R 6 6 6 0 −2 0 0 0 0 −2 −2 0 1 0 0 0 0 0
1920.240996.10a R 10 2 −2 4 2 0 1 2 −2 −2 0 0 0 1 1 −1 0 −1
1920.240996.10b R 10 2 −2 −2 −2 2 1 −4 0 2 0 0 0 1 1 −1 0 −1
1920.240996.10c R 10 2 −2 −4 2 0 1 −2 2 −2 0 0 0 1 −1 −1 0 1
1920.240996.10d R 10 2 −2 2 −2 −2 1 4 0 2 0 0 0 1 −1 −1 0 1
1920.240996.10e R 10 −6 2 0 2 0 −2 0 0 2 −2 0 0 2 0 0 0 0
1920.240996.15a R 15 −9 3 −3 −1 1 0 3 −1 −1 1 1 0 0 0 0 −1 0
1920.240996.15b R 15 −9 3 3 −1 −1 0 −3 1 −1 1 −1 0 0 0 0 1 0
1920.240996.20a R 20 4 −4 2 0 2 −1 −2 −2 0 0 0 0 −1 −1 1 0 1
1920.240996.20b R 20 4 −4 −2 0 −2 −1 2 2 0 0 0 0 −1 1 1 0 −1

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

$f_{ 1 } =$ $x^{10} + 25 t x^{6} - 3 x^{4} - 3 t$ Copy content Toggle raw display