Properties

Label 32T37
Degree $32$
Order $32$
Cyclic no
Abelian yes
Solvable yes
Transitivity $1$
Primitive no
$p$-group yes
Group: $C_2^2\times C_8$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(32, 37);
 
Copy content sage:G = TransitiveGroup(32, 37)
 
Copy content oscar:G = transitive_group(32, 37)
 
Copy content gap:G := TransitiveGroup(32, 37);
 

Group invariants

Abstract group:  $C_2^2\times C_8$
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:  yes
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:  $1$
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 $t$:  $37$
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:  $(1,13,12,23,2,14,11,24)(3,16,9,22,4,15,10,21)(5,20,29,28,6,19,30,27)(7,18,31,26,8,17,32,25)$, $(1,4)(2,3)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)$, $(1,19,2,20)(3,18,4,17)(5,24,6,23)(7,22,8,21)(9,26,10,25)(11,28,12,27)(13,30,14,29)(15,32,16,31)$
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
$8$:  $C_8$ x 4, $C_4\times C_2$ x 6, $C_2^3$
$16$:  $C_4\times C_2^2$, $C_8\times C_2$ x 6

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_4$ x 4, $C_2^2$ x 7

Degree 8: $C_8$ x 4, $C_4\times C_2$ x 6, $C_2^3$

Degree 16: $C_4\times C_2^2$, $C_8\times C_2$ x 6

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 $2^{16}$ $1$ $2$ $16$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)$
2B $2^{16}$ $1$ $2$ $16$ $( 1,28)( 2,27)( 3,25)( 4,26)( 5,14)( 6,13)( 7,16)( 8,15)( 9,18)(10,17)(11,20)(12,19)(21,32)(22,31)(23,30)(24,29)$
2C $2^{16}$ $1$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)$
2D $2^{16}$ $1$ $2$ $16$ $( 1,27)( 2,28)( 3,26)( 4,25)( 5,13)( 6,14)( 7,15)( 8,16)( 9,17)(10,18)(11,19)(12,20)(21,31)(22,32)(23,29)(24,30)$
2E $2^{16}$ $1$ $2$ $16$ $( 1,26)( 2,25)( 3,27)( 4,28)( 5,16)( 6,15)( 7,14)( 8,13)( 9,20)(10,19)(11,18)(12,17)(21,30)(22,29)(23,32)(24,31)$
2F $2^{16}$ $1$ $2$ $16$ $( 1,25)( 2,26)( 3,28)( 4,27)( 5,15)( 6,16)( 7,13)( 8,14)( 9,19)(10,20)(11,17)(12,18)(21,29)(22,30)(23,31)(24,32)$
2G $2^{16}$ $1$ $2$ $16$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
4A1 $4^{8}$ $1$ $4$ $24$ $( 1,10, 2, 9)( 3,11, 4,12)( 5,31, 6,32)( 7,29, 8,30)(13,21,14,22)(15,23,16,24)(17,27,18,28)(19,25,20,26)$
4A-1 $4^{8}$ $1$ $4$ $24$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,32, 6,31)( 7,30, 8,29)(13,22,14,21)(15,24,16,23)(17,28,18,27)(19,26,20,25)$
4B1 $4^{8}$ $1$ $4$ $24$ $( 1,19, 2,20)( 3,18, 4,17)( 5,24, 6,23)( 7,22, 8,21)( 9,26,10,25)(11,28,12,27)(13,30,14,29)(15,32,16,31)$
4B-1 $4^{8}$ $1$ $4$ $24$ $( 1,20, 2,19)( 3,17, 4,18)( 5,23, 6,24)( 7,21, 8,22)( 9,25,10,26)(11,27,12,28)(13,29,14,30)(15,31,16,32)$
4C1 $4^{8}$ $1$ $4$ $24$ $( 1,17, 2,18)( 3,20, 4,19)( 5,22, 6,21)( 7,24, 8,23)( 9,28,10,27)(11,26,12,25)(13,32,14,31)(15,30,16,29)$
4C-1 $4^{8}$ $1$ $4$ $24$ $( 1,18, 2,17)( 3,19, 4,20)( 5,21, 6,22)( 7,23, 8,24)( 9,27,10,28)(11,25,12,26)(13,31,14,32)(15,29,16,30)$
4D1 $4^{8}$ $1$ $4$ $24$ $( 1,12, 2,11)( 3, 9, 4,10)( 5,29, 6,30)( 7,31, 8,32)(13,23,14,24)(15,21,16,22)(17,25,18,26)(19,27,20,28)$
4D-1 $4^{8}$ $1$ $4$ $24$ $( 1,11, 2,12)( 3,10, 4, 9)( 5,30, 6,29)( 7,32, 8,31)(13,24,14,23)(15,22,16,21)(17,26,18,25)(19,28,20,27)$
8A1 $8^{4}$ $1$ $8$ $28$ $( 1,13,12,23, 2,14,11,24)( 3,16, 9,22, 4,15,10,21)( 5,20,29,28, 6,19,30,27)( 7,18,31,26, 8,17,32,25)$
8A-1 $8^{4}$ $1$ $8$ $28$ $( 1,24,11,14, 2,23,12,13)( 3,21,10,15, 4,22, 9,16)( 5,27,30,19, 6,28,29,20)( 7,25,32,17, 8,26,31,18)$
8A3 $8^{4}$ $1$ $8$ $28$ $( 1,23,11,13, 2,24,12,14)( 3,22,10,16, 4,21, 9,15)( 5,28,30,20, 6,27,29,19)( 7,26,32,18, 8,25,31,17)$
8A-3 $8^{4}$ $1$ $8$ $28$ $( 1,14,12,24, 2,13,11,23)( 3,15, 9,21, 4,16,10,22)( 5,19,29,27, 6,20,30,28)( 7,17,31,25, 8,18,32,26)$
8B1 $8^{4}$ $1$ $8$ $28$ $( 1,15,12,21, 2,16,11,22)( 3,14, 9,24, 4,13,10,23)( 5,18,29,26, 6,17,30,25)( 7,20,31,28, 8,19,32,27)$
8B-1 $8^{4}$ $1$ $8$ $28$ $( 1,22,11,16, 2,21,12,15)( 3,23,10,13, 4,24, 9,14)( 5,25,30,17, 6,26,29,18)( 7,27,32,19, 8,28,31,20)$
8B3 $8^{4}$ $1$ $8$ $28$ $( 1,21,11,15, 2,22,12,16)( 3,24,10,14, 4,23, 9,13)( 5,26,30,18, 6,25,29,17)( 7,28,32,20, 8,27,31,19)$
8B-3 $8^{4}$ $1$ $8$ $28$ $( 1,16,12,22, 2,15,11,21)( 3,13, 9,23, 4,14,10,24)( 5,17,29,25, 6,18,30,26)( 7,19,31,27, 8,20,32,28)$
8C1 $8^{4}$ $1$ $8$ $28$ $( 1, 6,12,30, 2, 5,11,29)( 3, 7, 9,31, 4, 8,10,32)(13,19,23,27,14,20,24,28)(15,17,21,25,16,18,22,26)$
8C-1 $8^{4}$ $1$ $8$ $28$ $( 1,29,11, 5, 2,30,12, 6)( 3,32,10, 8, 4,31, 9, 7)(13,28,24,20,14,27,23,19)(15,26,22,18,16,25,21,17)$
8C3 $8^{4}$ $1$ $8$ $28$ $( 1,30,11, 6, 2,29,12, 5)( 3,31,10, 7, 4,32, 9, 8)(13,27,24,19,14,28,23,20)(15,25,22,17,16,26,21,18)$
8C-3 $8^{4}$ $1$ $8$ $28$ $( 1, 5,12,29, 2, 6,11,30)( 3, 8, 9,32, 4, 7,10,31)(13,20,23,28,14,19,24,27)(15,18,21,26,16,17,22,25)$
8D1 $8^{4}$ $1$ $8$ $28$ $( 1, 8,12,32, 2, 7,11,31)( 3, 5, 9,29, 4, 6,10,30)(13,17,23,25,14,18,24,26)(15,19,21,27,16,20,22,28)$
8D-1 $8^{4}$ $1$ $8$ $28$ $( 1,31,11, 7, 2,32,12, 8)( 3,30,10, 6, 4,29, 9, 5)(13,26,24,18,14,25,23,17)(15,28,22,20,16,27,21,19)$
8D3 $8^{4}$ $1$ $8$ $28$ $( 1,32,11, 8, 2,31,12, 7)( 3,29,10, 5, 4,30, 9, 6)(13,25,24,17,14,26,23,18)(15,27,22,19,16,28,21,20)$
8D-3 $8^{4}$ $1$ $8$ $28$ $( 1, 7,12,31, 2, 8,11,32)( 3, 6, 9,30, 4, 5,10,29)(13,18,23,26,14,17,24,25)(15,20,21,28,16,19,22,27)$

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

32 x 32 character table

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