Properties

Label 32T49
Degree $32$
Order $32$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $C_2\times Q_{16}$

Related objects

Downloads

Learn more

Show commands: Gap / Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(32, 49);
 
Copy content sage:G = TransitiveGroup(32, 49)
 
Copy content oscar:G = transitive_group(32, 49)
 
Copy content gap:G := TransitiveGroup(32, 49);
 

Group invariants

Abstract group:  $C_2\times Q_{16}$
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:  $32=2^{5}$
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:  yes
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:  $3$
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 $n$:  $32$
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 $t$:  $49$
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);
 
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);
 
$\card{\Aut(F/K)}$:  $32$
Copy content comment:Order of the centralizer of G in S_n
 
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Copy content sage:SymmetricGroup(32).centralizer(G).order()
 
Copy content oscar:order(centralizer(symmetric_group(32), G)[1])
 
Copy content gap:Order(Centralizer(SymmetricGroup(32), G));
 
Generators:  $(1,30,2,29)(3,32,4,31)(5,10,6,9)(7,11,8,12)(13,15,14,16)(17,19,18,20)(21,23,22,24)(25,27,26,28)$, $(1,11,2,12)(3,9,4,10)(5,31,6,32)(7,30,8,29)(13,28,14,27)(15,25,16,26)(17,24,18,23)(19,21,20,22)$, $(1,23,2,24)(3,21,4,22)(5,17,6,18)(7,20,8,19)(9,13,10,14)(11,16,12,15)(25,32,26,31)(27,29,28,30)$
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 7
$4$:  $C_2^2$ x 7
$8$:  $D_{4}$ x 2, $C_2^3$
$16$:  $D_4\times C_2$, $Q_{16}$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7, $D_{4}$ x 4

Degree 8: $C_2^3$, $D_4$ x 2, $D_4\times C_2$ x 4

Degree 16: $D_4\times C_2$, $Q_{16}$ x 2

Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{32}$ $1$ $1$ $0$ $()$
2A $2^{16}$ $1$ $2$ $16$ $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,32)(10,31)(11,30)(12,29)(13,26)(14,25)(15,28)(16,27)(17,22)(18,21)(19,24)(20,23)$
2B $2^{16}$ $1$ $2$ $16$ $( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,31)(10,32)(11,29)(12,30)(13,25)(14,26)(15,27)(16,28)(17,21)(18,22)(19,23)(20,24)$
2C $2^{16}$ $1$ $2$ $16$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
4A $4^{8}$ $2$ $4$ $24$ $( 1,10, 2, 9)( 3,11, 4,12)( 5,29, 6,30)( 7,31, 8,32)(13,24,14,23)(15,22,16,21)(17,27,18,28)(19,25,20,26)$
4B $4^{8}$ $2$ $4$ $24$ $( 1,31, 2,32)( 3,30, 4,29)( 5,12, 6,11)( 7,10, 8, 9)(13,19,14,20)(15,17,16,18)(21,28,22,27)(23,26,24,25)$
4C $4^{8}$ $4$ $4$ $24$ $( 1,11, 2,12)( 3, 9, 4,10)( 5,31, 6,32)( 7,30, 8,29)(13,28,14,27)(15,25,16,26)(17,24,18,23)(19,21,20,22)$
4D $4^{8}$ $4$ $4$ $24$ $( 1,24, 2,23)( 3,22, 4,21)( 5,18, 6,17)( 7,19, 8,20)( 9,14,10,13)(11,15,12,16)(25,31,26,32)(27,30,28,29)$
4E $4^{8}$ $4$ $4$ $24$ $( 1,30, 2,29)( 3,32, 4,31)( 5,10, 6, 9)( 7,11, 8,12)(13,15,14,16)(17,19,18,20)(21,23,22,24)(25,27,26,28)$
4F $4^{8}$ $4$ $4$ $24$ $( 1,19, 2,20)( 3,17, 4,18)( 5,21, 6,22)( 7,24, 8,23)( 9,25,10,26)(11,28,12,27)(13,32,14,31)(15,29,16,30)$
8A1 $8^{4}$ $2$ $8$ $28$ $( 1,16,31,18, 2,15,32,17)( 3,13,30,19, 4,14,29,20)( 5,25,12,23, 6,26,11,24)( 7,27,10,21, 8,28, 9,22)$
8A3 $8^{4}$ $2$ $8$ $28$ $( 1,15,31,17, 2,16,32,18)( 3,14,30,20, 4,13,29,19)( 5,26,12,24, 6,25,11,23)( 7,28,10,22, 8,27, 9,21)$
8B1 $8^{4}$ $2$ $8$ $28$ $( 1,27,31,21, 2,28,32,22)( 3,26,30,24, 4,25,29,23)( 5,14,12,20, 6,13,11,19)( 7,16,10,18, 8,15, 9,17)$
8B3 $8^{4}$ $2$ $8$ $28$ $( 1,28,31,22, 2,27,32,21)( 3,25,30,23, 4,26,29,24)( 5,13,12,19, 6,14,11,20)( 7,15,10,17, 8,16, 9,18)$

Malle's constant $a(G)$:     $1/16$

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 4A 4B 4C 4D 4E 4F 8A1 8A3 8B1 8B3
Size 1 1 1 1 2 2 4 4 4 4 2 2 2 2
2 P 1A 1A 1A 1A 2C 2C 2C 2C 2C 2C 4B 4B 4B 4B
Type
32.41.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.41.1b R 1 −1 −1 1 −1 1 −1 −1 1 1 1 1 −1 −1
32.41.1c R 1 −1 −1 1 −1 1 −1 1 1 −1 −1 −1 1 1
32.41.1d R 1 −1 −1 1 −1 1 1 −1 −1 1 −1 −1 1 1
32.41.1e R 1 −1 −1 1 −1 1 1 1 −1 −1 1 1 −1 −1
32.41.1f R 1 1 1 1 1 1 −1 −1 −1 −1 1 1 1 1
32.41.1g R 1 1 1 1 1 1 −1 1 −1 1 −1 −1 −1 −1
32.41.1h R 1 1 1 1 1 1 1 −1 1 −1 −1 −1 −1 −1
32.41.2a R 2 −2 −2 2 2 −2 0 0 0 0 0 0 0 0
32.41.2b R 2 2 2 2 −2 −2 0 0 0 0 0 0 0 0
32.41.2c1 S 2 −2 2 −2 0 0 0 0 0 0 −ζ8−1−ζ8 ζ8−1+ζ8 ζ8−1+ζ8 −ζ8−1−ζ8
32.41.2c2 S 2 −2 2 −2 0 0 0 0 0 0 ζ8−1+ζ8 −ζ8−1−ζ8 −ζ8−1−ζ8 ζ8−1+ζ8
32.41.2d1 S 2 2 −2 −2 0 0 0 0 0 0 −ζ8−1−ζ8 ζ8−1+ζ8 −ζ8−1−ζ8 ζ8−1+ζ8
32.41.2d2 S 2 2 −2 −2 0 0 0 0 0 0 ζ8−1+ζ8 −ζ8−1−ζ8 ζ8−1+ζ8 −ζ8−1−ζ8

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

Data not computed