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Properties

Label 32T31
32T31 1 6 1->6 11 1->11 2 5 2->5 12 2->12 3 7 3->7 10 3->10 4 8 4->8 9 4->9 5->7 6->8 29 9->29 30 10->30 31 11->31 32 12->32 13 26 13->26 13->30 14 25 14->25 14->29 15 28 15->28 15->32 16 27 16->27 16->31 17 22 17->22 17->25 18 21 18->21 18->26 19 23 19->23 19->28 20 24 20->24 20->27 21->22 23->24
Degree $32$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $D_{16}$

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Copy content magma:G := TransitiveGroup(32, 31);
 

Group invariants

Abstract group:  $D_{16}$
Copy content magma:IdentifyGroup(G);
 
Order:  $32=2^{5}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  no
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:  $4$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $32$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $31$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $32$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,6)(2,5)(3,7)(4,8)(9,29)(10,30)(11,31)(12,32)(13,26)(14,25)(15,28)(16,27)(17,22)(18,21)(19,23)(20,24)$, $(1,11)(2,12)(3,10)(4,9)(5,7)(6,8)(13,30)(14,29)(15,32)(16,31)(17,25)(18,26)(19,28)(20,27)(21,22)(23,24)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $D_{4}$
$16$:  $D_{8}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4$, $D_{8}$ x 2

Degree 16: $D_{8}$, $D_{16}$ x 2

Low degree siblings

16T56 x 2

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 $2^{16}$ $1$ $2$ $16$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,19)(18,20)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)$
2B $2^{16}$ $8$ $2$ $16$ $( 1,25)( 2,26)( 3,28)( 4,27)( 5,23)( 6,24)( 7,21)( 8,22)( 9,18)(10,17)(11,19)(12,20)(13,14)(15,16)(29,31)(30,32)$
2C $2^{16}$ $8$ $2$ $16$ $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,32)(10,31)(11,30)(12,29)(13,27)(14,28)(15,25)(16,26)(17,23)(18,24)(19,22)(20,21)$
4A $4^{8}$ $2$ $4$ $24$ $( 1,19, 3,17)( 2,20, 4,18)( 5,21, 8,24)( 6,22, 7,23)( 9,27,12,26)(10,28,11,25)(13,32,16,29)(14,31,15,30)$
8A1 $8^{4}$ $2$ $8$ $28$ $( 1,27,19,12, 3,26,17, 9)( 2,28,20,11, 4,25,18,10)( 5,30,21,14, 8,31,24,15)( 6,29,22,13, 7,32,23,16)$
8A3 $8^{4}$ $2$ $8$ $28$ $( 1,12,17,27, 3, 9,19,26)( 2,11,18,28, 4,10,20,25)( 5,14,24,30, 8,15,21,31)( 6,13,23,29, 7,16,22,32)$
16A1 $16^{2}$ $2$ $16$ $30$ $( 1,31,27,24,19,15,12, 5, 3,30,26,21,17,14, 9, 8)( 2,32,28,23,20,16,11, 6, 4,29,25,22,18,13,10, 7)$
16A3 $16^{2}$ $2$ $16$ $30$ $( 1,24,12,30,17, 8,27,15, 3,21, 9,31,19, 5,26,14)( 2,23,11,29,18, 7,28,16, 4,22,10,32,20, 6,25,13)$
16A5 $16^{2}$ $2$ $16$ $30$ $( 1,15,26, 8,19,30, 9,24, 3,14,27, 5,17,31,12,21)( 2,16,25, 7,20,29,10,23, 4,13,28, 6,18,32,11,22)$
16A7 $16^{2}$ $2$ $16$ $30$ $( 1,30,27,21,19,14,12, 8, 3,31,26,24,17,15, 9, 5)( 2,29,28,22,20,13,11, 7, 4,32,25,23,18,16,10, 6)$

Malle's constant $a(G)$:     $1/16$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 4A 8A1 8A3 16A1 16A3 16A5 16A7
Size 1 1 8 8 2 2 2 2 2 2 2
2 P 1A 1A 1A 1A 2A 4A 4A 8A1 8A3 8A3 8A1
Type
32.18.1a R 1 1 1 1 1 1 1 1 1 1 1
32.18.1b R 1 1 −1 −1 1 1 1 1 1 1 1
32.18.1c R 1 1 −1 1 1 1 1 −1 −1 −1 −1
32.18.1d R 1 1 1 −1 1 1 1 −1 −1 −1 −1
32.18.2a R 2 2 0 0 2 −2 −2 0 0 0 0
32.18.2b1 R 2 2 0 0 −2 0 0 −ζ8−1−ζ8 ζ8−1+ζ8 ζ8−1+ζ8 −ζ8−1−ζ8
32.18.2b2 R 2 2 0 0 −2 0 0 ζ8−1+ζ8 −ζ8−1−ζ8 −ζ8−1−ζ8 ζ8−1+ζ8
32.18.2c1 R 2 −2 0 0 0 −ζ16−2−ζ162 ζ16−2+ζ162 −ζ16−3−ζ163 ζ16−1+ζ16 −ζ16−1−ζ16 ζ16−3+ζ163
32.18.2c2 R 2 −2 0 0 0 −ζ16−2−ζ162 ζ16−2+ζ162 ζ16−3+ζ163 −ζ16−1−ζ16 ζ16−1+ζ16 −ζ16−3−ζ163
32.18.2c3 R 2 −2 0 0 0 ζ16−2+ζ162 −ζ16−2−ζ162 −ζ16−1−ζ16 −ζ16−3−ζ163 ζ16−3+ζ163 ζ16−1+ζ16
32.18.2c4 R 2 −2 0 0 0 ζ16−2+ζ162 −ζ16−2−ζ162 ζ16−1+ζ16 ζ16−3+ζ163 −ζ16−3−ζ163 −ζ16−1−ζ16

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed