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Properties

Label 32T45
Degree $32$
Order $32$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $C_4:Q_8$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(32, 45);
 
Copy content sage:G = TransitiveGroup(32, 45)
 
Copy content oscar:G = transitive_group(32, 45)
 
Copy content gap:G := TransitiveGroup(32, 45);
 

Group invariants

Abstract group:  $C_4:Q_8$
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:  $2$
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 $t$:  $45$
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:  $(1,30,3,32)(2,29,4,31)(5,10,7,12)(6,9,8,11)(13,20,15,18)(14,19,16,17)(21,28,23,26)(22,27,24,25)$, $(1,22,3,24)(2,21,4,23)(5,20,7,18)(6,19,8,17)(9,14,11,16)(10,13,12,15)(25,32,27,30)(26,31,28,29)$, $(1,8,3,6)(2,7,4,5)(9,29,11,31)(10,30,12,32)(13,27,15,25)(14,28,16,26)(17,22,19,24)(18,21,20,23)$
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, $C_2^3$, $Q_8$ x 4
$16$:  $D_4\times C_2$, $Q_8\times C_2$ 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, $Q_8$ x 4, $D_4\times C_2$ x 4

Degree 16: $Q_8\times C_2$ x 2, $D_4\times C_2$

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

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 4G 4H 4I 4J
Size 1 1 1 1 2 2 2 2 2 2 4 4 4 4
2 P 1A 1A 1A 1A 2A 2B 2A 2B 2C 2C 2C 2C 2C 2C
Type
32.35.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.35.1b R 1 1 1 1 −1 −1 −1 −1 1 1 −1 1 1 −1
32.35.1c R 1 1 1 1 −1 −1 −1 −1 1 1 1 −1 −1 1
32.35.1d R 1 1 1 1 −1 1 −1 1 −1 −1 −1 1 −1 1
32.35.1e R 1 1 1 1 −1 1 −1 1 −1 −1 1 −1 1 −1
32.35.1f R 1 1 1 1 1 −1 1 −1 −1 −1 −1 −1 1 1
32.35.1g R 1 1 1 1 1 −1 1 −1 −1 −1 1 1 −1 −1
32.35.1h R 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
32.35.2a R 2 −2 −2 2 0 0 0 0 −2 2 0 0 0 0
32.35.2b R 2 −2 −2 2 0 0 0 0 2 −2 0 0 0 0
32.35.2c S 2 −2 2 −2 0 −2 0 2 0 0 0 0 0 0
32.35.2d S 2 −2 2 −2 0 2 0 −2 0 0 0 0 0 0
32.35.2e S 2 2 −2 −2 −2 0 2 0 0 0 0 0 0 0
32.35.2f S 2 2 −2 −2 2 0 −2 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

Data not computed