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Properties

Label 32T1979
32T1979 1 8 1->8 17 1->17 2 7 2->7 18 2->18 3 5 3->5 11 3->11 4 6 4->6 12 4->12 15 5->15 20 5->20 16 6->16 19 6->19 10 7->10 22 7->22 9 8->9 21 8->21 26 9->26 31 9->31 25 10->25 32 10->32 11->21 12->22 13 13->3 29 13->29 14 14->4 30 14->30 15->6 15->26 16->5 16->25 17->15 18->16 27 19->27 19->29 28 20->28 20->30 21->7 21->28 22->8 22->27 23 23->2 23->32 24 24->1 24->31 25->13 26->14 27->23 28->24 29->14 29->18 30->13 30->17 31->12 31->23 32->11 32->24
Degree $32$
Order $128$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $(C_2\times C_8).D_4$

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Copy content comment:Define the Galois group
 
Copy content magma:G := TransitiveGroup(32, 1979);
 
Copy content sage:G = TransitiveGroup(32, 1979)
 
Copy content oscar:G = transitive_group(32, 1979)
 

Group invariants

Abstract group:  $(C_2\times C_8).D_4$
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 $t$:  $1979$
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)
 
$\card{\Aut(F/K)}$:  $8$
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:  $(1,17)(2,18)(3,11)(4,12)(5,15,6,16)(7,22,8,21)(9,26)(10,25)(13,29,14,30)(19,27)(20,28)(23,32,24,31)$, $(1,8,9,31,12,22,27,23,2,7,10,32,11,21,28,24)(3,5,20,30,17,15,26,14,4,6,19,29,18,16,25,13)$
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, $C_4\times C_2$
$16$:  $C_8:C_2$, $C_2^2:C_4$, $C_8\times C_2$
$32$:  $C_2^3 : C_4 $, $C_2^2 : C_8$, $C_4.D_4$
$64$:  32T348

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$, $D_{4}$ x 2

Degree 8: $C_8$, $C_8:C_2$, $C_2^2:C_4$

Degree 16: $C_2^2 : C_8$

Low degree siblings

32T1979, 32T2101

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

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

1A 2A 2B 2C 4A1 4A-1 4B 4C 4D 4E 8A1 8A-1 8B1 8B-1 8C1 8C-1 8C3 8C-3 16A1 16A-1 16A3 16A-3 16B1 16B-1 16B3 16B-3
Size 1 1 2 4 1 1 2 4 8 8 4 4 4 4 4 4 4 4 8 8 8 8 8 8 8 8
2 P 1A 1A 1A 1A 2A 2A 2A 2A 2B 2B 4A1 4A-1 4A-1 4A1 4A1 4A-1 4A-1 4A1 8A1 8A-1 8A-1 8A1 8B1 8B-1 8B-1 8B1
Type
128.60.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
128.60.1b R 1 1 1 1 1 1 1 1 −1 −1 1 −1 1 1 1 −1 −1 −1 −1 −1 1 1 1 −1 −1 1
128.60.1c R 1 1 1 1 1 1 1 1 −1 −1 1 −1 1 1 1 −1 −1 −1 1 1 −1 −1 −1 1 1 −1
128.60.1d R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
128.60.1e1 C 1 1 1 1 1 1 1 1 −1 −1 −1 1 −1 −1 −1 1 1 1 −i −i i i −i i i −i
128.60.1e2 C 1 1 1 1 1 1 1 1 −1 −1 −1 1 −1 −1 −1 1 1 1 i i −i −i i −i −i i
128.60.1f1 C 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 −i −i −i −i i i i i
128.60.1f2 C 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1 i i i i −i −i −i −i
128.60.1g1 C 1 1 1 1 −1 −1 −1 −1 −1 1 −ζ82 ζ82 ζ82 −ζ82 ζ82 −ζ82 ζ82 −ζ82 −ζ8 ζ8 ζ8 −ζ8 −ζ83 ζ83 −ζ83 ζ83
128.60.1g2 C 1 1 1 1 −1 −1 −1 −1 −1 1 ζ82 −ζ82 −ζ82 ζ82 −ζ82 ζ82 −ζ82 ζ82 ζ83 −ζ83 −ζ83 ζ83 ζ8 −ζ8 ζ8 −ζ8
128.60.1g3 C 1 1 1 1 −1 −1 −1 −1 −1 1 −ζ82 ζ82 ζ82 −ζ82 ζ82 −ζ82 ζ82 −ζ82 ζ8 −ζ8 −ζ8 ζ8 ζ83 −ζ83 ζ83 −ζ83
128.60.1g4 C 1 1 1 1 −1 −1 −1 −1 −1 1 ζ82 −ζ82 −ζ82 ζ82 −ζ82 ζ82 −ζ82 ζ82 −ζ83 ζ83 ζ83 −ζ83 −ζ8 ζ8 −ζ8 ζ8
128.60.1h1 C 1 1 1 1 −1 −1 −1 −1 1 −1 −ζ82 −ζ82 ζ82 −ζ82 ζ82 ζ82 −ζ82 ζ82 −ζ8 ζ8 −ζ8 ζ8 ζ83 ζ83 −ζ83 −ζ83
128.60.1h2 C 1 1 1 1 −1 −1 −1 −1 1 −1 ζ82 ζ82 −ζ82 ζ82 −ζ82 −ζ82 ζ82 −ζ82 ζ83 −ζ83 ζ83 −ζ83 −ζ8 −ζ8 ζ8 ζ8
128.60.1h3 C 1 1 1 1 −1 −1 −1 −1 1 −1 −ζ82 −ζ82 ζ82 −ζ82 ζ82 ζ82 −ζ82 ζ82 ζ8 −ζ8 ζ8 −ζ8 −ζ83 −ζ83 ζ83 ζ83
128.60.1h4 C 1 1 1 1 −1 −1 −1 −1 1 −1 ζ82 ζ82 −ζ82 ζ82 −ζ82 −ζ82 ζ82 −ζ82 −ζ83 ζ83 −ζ83 ζ83 ζ8 ζ8 −ζ8 −ζ8
128.60.2a R 2 2 2 −2 2 2 2 −2 0 0 −2 0 −2 2 2 0 0 0 0 0 0 0 0 0 0 0
128.60.2b R 2 2 2 −2 2 2 2 −2 0 0 2 0 2 −2 −2 0 0 0 0 0 0 0 0 0 0 0
128.60.2c1 C 2 2 2 −2 −2 −2 −2 2 0 0 −2i 0 2i 2i −2i 0 0 0 0 0 0 0 0 0 0 0
128.60.2c2 C 2 2 2 −2 −2 −2 −2 2 0 0 2i 0 −2i −2i 2i 0 0 0 0 0 0 0 0 0 0 0
128.60.4a R 4 4 −4 0 4 4 −4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
128.60.4b S 4 4 −4 0 −4 −4 4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
128.60.4c1 C 4 −4 0 0 −4ζ82 4ζ82 0 0 0 0 0 −2ζ8 0 0 0 2ζ83 2ζ8 −2ζ83 0 0 0 0 0 0 0 0
128.60.4c2 C 4 −4 0 0 4ζ82 −4ζ82 0 0 0 0 0 2ζ83 0 0 0 −2ζ8 −2ζ83 2ζ8 0 0 0 0 0 0 0 0
128.60.4c3 C 4 −4 0 0 −4ζ82 4ζ82 0 0 0 0 0 2ζ8 0 0 0 −2ζ83 −2ζ8 2ζ83 0 0 0 0 0 0 0 0
128.60.4c4 C 4 −4 0 0 4ζ82 −4ζ82 0 0 0 0 0 −2ζ83 0 0 0 2ζ8 2ζ83 −2ζ8 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)
 

Regular extensions

Data not computed