Properties

Label 32T5
Degree $32$
Order $32$
Cyclic no
Abelian no
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $C_4\times D_4$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(32, 5);
 
Copy content sage:G = TransitiveGroup(32, 5)
 
Copy content oscar:G = transitive_group(32, 5)
 
Copy content gap:G := TransitiveGroup(32, 5);
 

Group invariants

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_4$ x 4, $C_2^2$ x 7, $D_{4}$ x 4

Degree 8: $C_4\times C_2$ x 6, $C_2^3$, $D_4$ x 2, $D_4\times C_2$ x 4, $Q_8:C_2$ x 3

Degree 16: $C_4\times C_2^2$, $D_4\times C_2$, $Q_8 : C_2$, $C_4 \times D_4$ x 4

Low degree siblings

16T19 x 4

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{32}$ $1$ $1$ $0$ $()$
2A $2^{16}$ $1$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,31)(30,32)$
2B $2^{16}$ $1$ $2$ $16$ $( 1,29)( 2,30)( 3,31)( 4,32)( 5,11)( 6,12)( 7,10)( 8, 9)(13,19)(14,20)(15,17)(16,18)(21,28)(22,27)(23,26)(24,25)$
2C $2^{16}$ $1$ $2$ $16$ $( 1,31)( 2,32)( 3,29)( 4,30)( 5,10)( 6, 9)( 7,11)( 8,12)(13,18)(14,17)(15,20)(16,19)(21,25)(22,26)(23,27)(24,28)$
2D $2^{16}$ $2$ $2$ $16$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,20)(14,19)(15,18)(16,17)(21,27)(22,28)(23,25)(24,26)(29,30)(31,32)$
2E $2^{16}$ $2$ $2$ $16$ $( 1,15)( 2,16)( 3,14)( 4,13)( 5,26)( 6,25)( 7,27)( 8,28)( 9,21)(10,22)(11,23)(12,24)(17,29)(18,30)(19,32)(20,31)$
2F $2^{16}$ $2$ $2$ $16$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,17)(14,18)(15,19)(16,20)(21,26)(22,25)(23,28)(24,27)(29,32)(30,31)$
2G $2^{16}$ $2$ $2$ $16$ $( 1,14)( 2,13)( 3,15)( 4,16)( 5,27)( 6,28)( 7,26)( 8,25)( 9,24)(10,23)(11,22)(12,21)(17,31)(18,32)(19,30)(20,29)$
4A1 $4^{8}$ $1$ $4$ $24$ $( 1, 7, 3, 5)( 2, 8, 4, 6)( 9,32,12,30)(10,31,11,29)(13,25,16,28)(14,26,15,27)(17,22,20,23)(18,21,19,24)$
4A-1 $4^{8}$ $1$ $4$ $24$ $( 1, 5, 3, 7)( 2, 6, 4, 8)( 9,30,12,32)(10,29,11,31)(13,28,16,25)(14,27,15,26)(17,23,20,22)(18,24,19,21)$
4B1 $4^{8}$ $1$ $4$ $24$ $( 1,10, 3,11)( 2, 9, 4,12)( 5,29, 7,31)( 6,30, 8,32)(13,24,16,21)(14,23,15,22)(17,27,20,26)(18,28,19,25)$
4B-1 $4^{8}$ $1$ $4$ $24$ $( 1,11, 3,10)( 2,12, 4, 9)( 5,31, 7,29)( 6,32, 8,30)(13,21,16,24)(14,22,15,23)(17,26,20,27)(18,25,19,28)$
4C $4^{8}$ $2$ $4$ $24$ $( 1,16,29,18)( 2,15,30,17)( 3,13,31,19)( 4,14,32,20)( 5,25,11,24)( 6,26,12,23)( 7,28,10,21)( 8,27, 9,22)$
4D $4^{8}$ $2$ $4$ $24$ $( 1,13,29,19)( 2,14,30,20)( 3,16,31,18)( 4,15,32,17)( 5,28,11,21)( 6,27,12,22)( 7,25,10,24)( 8,26, 9,23)$
4E1 $4^{8}$ $2$ $4$ $24$ $( 1,28,31,24)( 2,27,32,23)( 3,25,29,21)( 4,26,30,22)( 5,16,10,19)( 6,15, 9,20)( 7,13,11,18)( 8,14,12,17)$
4E-1 $4^{8}$ $2$ $4$ $24$ $( 1,25,31,21)( 2,26,32,22)( 3,28,29,24)( 4,27,30,23)( 5,13,10,18)( 6,14, 9,17)( 7,16,11,19)( 8,15,12,20)$
4F1 $4^{8}$ $2$ $4$ $24$ $( 1, 8, 3, 6)( 2, 7, 4, 5)( 9,31,12,29)(10,32,11,30)(13,23,16,22)(14,24,15,21)(17,28,20,25)(18,27,19,26)$
4F-1 $4^{8}$ $2$ $4$ $24$ $( 1, 6, 3, 8)( 2, 5, 4, 7)( 9,29,12,31)(10,30,11,32)(13,22,16,23)(14,21,15,24)(17,25,20,28)(18,26,19,27)$
4G1 $4^{8}$ $2$ $4$ $24$ $( 1,27, 3,26)( 2,28, 4,25)( 5,15, 7,14)( 6,16, 8,13)( 9,19,12,18)(10,20,11,17)(21,32,24,30)(22,31,23,29)$
4G-1 $4^{8}$ $2$ $4$ $24$ $( 1,26, 3,27)( 2,25, 4,28)( 5,14, 7,15)( 6,13, 8,16)( 9,18,12,19)(10,17,11,20)(21,30,24,32)(22,29,23,31)$

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 2D 2E 2F 2G 4A1 4A-1 4B1 4B-1 4C 4D 4E1 4E-1 4F1 4F-1 4G1 4G-1
Size 1 1 1 1 2 2 2 2 1 1 1 1 2 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2B 2B 2C 2C 2A 2A 2A 2A
Type
32.25.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.25.1b R 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 1 1 −1 −1 1 1 1 1
32.25.1c R 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
32.25.1d R 1 1 1 1 −1 1 −1 1 −1 −1 −1 −1 −1 −1 1 1 1 1 −1 −1
32.25.1e R 1 1 1 1 −1 1 −1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 1 1
32.25.1f R 1 1 1 1 1 −1 1 −1 −1 −1 −1 −1 −1 −1 1 1 −1 −1 1 1
32.25.1g R 1 1 1 1 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1
32.25.1h R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1 −1 −1 −1 −1
32.25.1i1 C 1 −1 1 −1 −1 −1 1 1 −i i −i i 1 −1 −i i i −i i −i
32.25.1i2 C 1 −1 1 −1 −1 −1 1 1 i −i i −i 1 −1 i −i −i i −i i
32.25.1j1 C 1 −1 1 −1 −1 1 1 −1 −i i −i i −1 1 i −i i −i −i i
32.25.1j2 C 1 −1 1 −1 −1 1 1 −1 i −i i −i −1 1 −i i −i i i −i
32.25.1k1 C 1 −1 1 −1 1 −1 −1 1 −i i −i i −1 1 i −i −i i i −i
32.25.1k2 C 1 −1 1 −1 1 −1 −1 1 i −i i −i −1 1 −i i i −i −i i
32.25.1l1 C 1 −1 1 −1 1 1 −1 −1 −i i −i i 1 −1 −i i −i i −i i
32.25.1l2 C 1 −1 1 −1 1 1 −1 −1 i −i i −i 1 −1 i −i i −i i −i
32.25.2a R 2 2 −2 −2 0 0 0 0 −2 −2 2 2 0 0 0 0 0 0 0 0
32.25.2b R 2 2 −2 −2 0 0 0 0 2 2 −2 −2 0 0 0 0 0 0 0 0
32.25.2c1 C 2 −2 −2 2 0 0 0 0 −2i 2i 2i −2i 0 0 0 0 0 0 0 0
32.25.2c2 C 2 −2 −2 2 0 0 0 0 2i −2i −2i 2i 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

Data not computed