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Properties

Label 32T600
32T600 1 25 1->25 26 1->26 2 2->25 2->26 3 27 3->27 28 3->28 4 4->27 4->28 5 30 5->30 32 5->32 6 29 6->29 31 6->31 7 7->30 7->32 8 8->29 8->31 9 19 9->19 20 9->20 10 10->19 10->20 11 17 11->17 18 11->18 12 12->17 12->18 13 22 13->22 24 13->24 14 21 14->21 23 14->23 15 15->22 15->24 16 16->21 16->23 17->10 17->14 18->9 18->13 19->11 19->16 20->12 20->15 21->1 21->13 22->2 22->14 23->3 23->15 24->4 24->16 25->4 25->6 26->3 26->5 27->2 27->8 28->1 28->7 29->7 29->11 30->8 30->12 31->5 31->10 32->6 32->9
Degree $32$
Order $128$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $D_8:C_8$

Related objects

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Show commands: Magma / Oscar / SageMath

Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(32, 600);
 
Copy content sage:G = TransitiveGroup(32, 600)
 
Copy content oscar:G = transitive_group(32, 600)
 

Group invariants

Abstract group:  $D_8:C_8$
Copy content comment:Abstract group ID
 
Copy content magma:IdentifyGroup(G);
 
Copy content sage:G.id()
 
Order:  $128=2^{7}$
Copy content comment:Order
 
Copy content magma:Order(G);
 
Copy content sage:G.order()
 
Copy content oscar: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)
 
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)
 
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)
 
Nilpotency class:  $4$
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
 

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)
 
Transitive number $t$:  $600$
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]
 
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)
 
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)
 
$\card{\Aut(F/K)}$:  $16$
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])
 
Generators:  $(1,26,5,32,9,20,15,24,4,27,8,29,11,17,14,21)(2,25,6,31,10,19,16,23,3,28,7,30,12,18,13,22)$, $(1,25,4,28)(2,26,3,27)(5,30,8,31)(6,29,7,32)(9,19,11,18)(10,20,12,17)(13,24,16,21)(14,23,15,22)$
Copy content comment:Generators
 
Copy content magma:Generators(G);
 
Copy content sage:G.gens()
 
Copy content oscar:gens(G)
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_8$ x 2, $C_4\times C_2$
$16$:  $D_{8}$, $C_8:C_2$, $QD_{16}$, $C_2^2:C_4$, $C_8\times C_2$
$32$:  $C_4\wr C_2$, $C_2^2 : C_8$, 16T26
$64$:  32T272

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4$, $C_4\wr C_2$ x 2

Degree 16: 16T42, $D_8:C_8$

Low degree siblings

16T260 x 2, 32T600

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 $2^{16}$ $1$ $2$ $16$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)$
2B $2^{8},1^{16}$ $2$ $2$ $8$ $(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)$
2C $2^{16}$ $8$ $2$ $16$ $( 1,18)( 2,17)( 3,20)( 4,19)( 5,23)( 6,24)( 7,21)( 8,22)( 9,25)(10,26)(11,28)(12,27)(13,32)(14,31)(15,30)(16,29)$
4A1 $4^{8}$ $1$ $4$ $24$ $( 1, 9, 4,11)( 2,10, 3,12)( 5,15, 8,14)( 6,16, 7,13)(17,26,20,27)(18,25,19,28)(21,32,24,29)(22,31,23,30)$
4A-1 $4^{8}$ $1$ $4$ $24$ $( 1,11, 4, 9)( 2,12, 3,10)( 5,14, 8,15)( 6,13, 7,16)(17,27,20,26)(18,28,19,25)(21,29,24,32)(22,30,23,31)$
4B $4^{8}$ $2$ $4$ $24$ $( 1, 9, 4,11)( 2,10, 3,12)( 5,15, 8,14)( 6,16, 7,13)(17,27,20,26)(18,28,19,25)(21,29,24,32)(22,30,23,31)$
4C1 $4^{4},2^{8}$ $2$ $4$ $20$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)(17,26,20,27)(18,25,19,28)(21,32,24,29)(22,31,23,30)$
4C-1 $4^{4},2^{8}$ $2$ $4$ $20$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)(17,27,20,26)(18,28,19,25)(21,29,24,32)(22,30,23,31)$
4D1 $4^{4},1^{16}$ $2$ $4$ $12$ $(17,27,20,26)(18,28,19,25)(21,29,24,32)(22,30,23,31)$
4D-1 $4^{4},1^{16}$ $2$ $4$ $12$ $(17,26,20,27)(18,25,19,28)(21,32,24,29)(22,31,23,30)$
4E $4^{8}$ $8$ $4$ $24$ $( 1,25, 4,28)( 2,26, 3,27)( 5,30, 8,31)( 6,29, 7,32)( 9,19,11,18)(10,20,12,17)(13,24,16,21)(14,23,15,22)$
8A1 $8^{4}$ $2$ $8$ $28$ $( 1, 5, 9,15, 4, 8,11,14)( 2, 6,10,16, 3, 7,12,13)(17,24,26,29,20,21,27,32)(18,23,25,30,19,22,28,31)$
8A-1 $8^{4}$ $2$ $8$ $28$ $( 1,15,11, 5, 4,14, 9, 8)( 2,16,12, 6, 3,13,10, 7)(17,29,27,24,20,32,26,21)(18,30,28,23,19,31,25,22)$
8B1 $8^{4}$ $2$ $8$ $28$ $( 1, 8, 9,14, 4, 5,11,15)( 2, 7,10,13, 3, 6,12,16)(17,24,26,29,20,21,27,32)(18,23,25,30,19,22,28,31)$
8B-1 $8^{4}$ $2$ $8$ $28$ $( 1,15,11, 5, 4,14, 9, 8)( 2,16,12, 6, 3,13,10, 7)(17,32,27,21,20,29,26,24)(18,31,28,22,19,30,25,23)$
8C $8^{4}$ $4$ $8$ $28$ $( 1, 8, 9,14, 4, 5,11,15)( 2, 7,10,13, 3, 6,12,16)(17,29,27,24,20,32,26,21)(18,30,28,23,19,31,25,22)$
8D $8^{4}$ $4$ $8$ $28$ $( 1,14,11, 8, 4,15, 9, 5)( 2,13,12, 7, 3,16,10, 6)(17,21,26,32,20,24,27,29)(18,22,25,31,19,23,28,30)$
8E1 $8^{2},4^{4}$ $4$ $8$ $26$ $( 1,10, 4,12)( 2, 9, 3,11)( 5,13, 8,16)( 6,14, 7,15)(17,31,26,23,20,30,27,22)(18,32,25,24,19,29,28,21)$
8E-1 $8^{2},4^{4}$ $4$ $8$ $26$ $( 1,12, 4,10)( 2,11, 3, 9)( 5,16, 8,13)( 6,15, 7,14)(17,22,27,30,20,23,26,31)(18,21,28,29,19,24,25,32)$
8E3 $8^{2},4^{4}$ $4$ $8$ $26$ $( 1,12, 4,10)( 2,11, 3, 9)( 5,16, 8,13)( 6,15, 7,14)(17,23,27,31,20,22,26,30)(18,24,28,32,19,21,25,29)$
8E-3 $8^{2},4^{4}$ $4$ $8$ $26$ $( 1,10, 4,12)( 2, 9, 3,11)( 5,13, 8,16)( 6,14, 7,15)(17,30,26,22,20,31,27,23)(18,29,25,21,19,32,28,24)$
8F1 $8^{2},2^{8}$ $4$ $8$ $22$ $( 1, 3)( 2, 4)( 5, 6)( 7, 8)( 9,12)(10,11)(13,14)(15,16)(17,23,27,31,20,22,26,30)(18,24,28,32,19,21,25,29)$
8F-1 $8^{2},2^{8}$ $4$ $8$ $22$ $( 1,16, 9, 7, 4,13,11, 6)( 2,15,10, 8, 3,14,12, 5)(17,19)(18,20)(21,22)(23,24)(25,27)(26,28)(29,30)(31,32)$
8F3 $8^{2},2^{8}$ $4$ $8$ $22$ $( 1,16, 9, 7, 4,13,11, 6)( 2,15,10, 8, 3,14,12, 5)(17,18)(19,20)(21,23)(22,24)(25,26)(27,28)(29,31)(30,32)$
8F-3 $8^{2},2^{8}$ $4$ $8$ $22$ $( 1, 3)( 2, 4)( 5, 6)( 7, 8)( 9,12)(10,11)(13,14)(15,16)(17,22,27,30,20,23,26,31)(18,21,28,29,19,24,25,32)$
8G1 $8^{4}$ $8$ $8$ $28$ $( 1,23, 9,30, 4,22,11,31)( 2,24,10,29, 3,21,12,32)( 5,25,15,19, 8,28,14,18)( 6,26,16,20, 7,27,13,17)$
8G-1 $8^{4}$ $8$ $8$ $28$ $( 1,25,11,18, 4,28, 9,19)( 2,26,12,17, 3,27,10,20)( 5,30,14,23, 8,31,15,22)( 6,29,13,24, 7,32,16,21)$
16A1 $16^{2}$ $8$ $16$ $30$ $( 1,32, 8,17, 9,24,14,26, 4,29, 5,20,11,21,15,27)( 2,31, 7,18,10,23,13,25, 3,30, 6,19,12,22,16,28)$
16A-1 $16^{2}$ $8$ $16$ $30$ $( 1,17,15,32,11,27, 5,21, 4,20,14,29, 9,26, 8,24)( 2,18,16,31,12,28, 6,22, 3,19,13,30,10,25, 7,23)$
16A3 $16^{2}$ $8$ $16$ $30$ $( 1,24,15,17,11,32, 5,27, 4,21,14,20, 9,29, 8,26)( 2,23,16,18,12,31, 6,28, 3,22,13,19,10,30, 7,25)$
16A-3 $16^{2}$ $8$ $16$ $30$ $( 1,24, 5,27, 9,29,15,17, 4,21, 8,26,11,32,14,20)( 2,23, 6,28,10,30,16,18, 3,22, 7,25,12,31,13,19)$

Malle's constant $a(G)$:     $1/8$

Copy content comment:Conjugacy classes
 
Copy content magma:ConjugacyClasses(G);
 
Copy content sage:G.conjugacy_classes()
 
Copy content oscar:conjugacy_classes(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)
 

Regular extensions

Data not computed