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Properties

Label 16T56
16T56 1 8 1->8 15 1->15 2 7 2->7 16 2->16 3 3->1 5 3->5 4 4->2 6 4->6 5->3 6->4 7->6 8->5 9 9->8 9->15 10 10->7 10->16 11 11->10 13 11->13 12 12->9 14 12->14 13->12 14->11 15->14 16->13
Degree $16$
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(16, 56);
 

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 $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $56$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $-1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $2$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,8)(2,7)(3,5)(4,6)(9,15)(10,16)(11,13)(12,14)$, $(1,15,14,11,10,7,6,4,2,16,13,12,9,8,5,3)$
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: $C_2$

Degree 4: $D_{4}$

Degree 8: $D_{8}$

Low degree siblings

16T56, 32T31

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

LabelCycle TypeSizeOrderIndexRepresentative
1A $1^{16}$ $1$ $1$ $0$ $()$
2A $2^{8}$ $1$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2B $2^{7},1^{2}$ $8$ $2$ $7$ $( 1,10)( 2, 9)( 3, 7)( 4, 8)( 5, 6)(11,15)(12,16)$
2C $2^{8}$ $8$ $2$ $8$ $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,16)(10,15)(11,14)(12,13)$
4A $4^{4}$ $2$ $4$ $12$ $( 1,10, 2, 9)( 3,11, 4,12)( 5,14, 6,13)( 7,16, 8,15)$
8A1 $8^{2}$ $2$ $8$ $14$ $( 1,13,10, 5, 2,14, 9, 6)( 3,16,11, 8, 4,15,12, 7)$
8A3 $8^{2}$ $2$ $8$ $14$ $( 1, 5, 9,13, 2, 6,10,14)( 3, 8,12,16, 4, 7,11,15)$
16A1 $16$ $2$ $16$ $15$ $( 1, 8,13, 4,10,15, 5,12, 2, 7,14, 3, 9,16, 6,11)$
16A3 $16$ $2$ $16$ $15$ $( 1, 4, 5, 7, 9,11,13,15, 2, 3, 6, 8,10,12,14,16)$
16A5 $16$ $2$ $16$ $15$ $( 1,15,14,11,10, 7, 6, 4, 2,16,13,12, 9, 8, 5, 3)$
16A7 $16$ $2$ $16$ $15$ $( 1, 7,13, 3,10,16, 5,11, 2, 8,14, 4, 9,15, 6,12)$

Malle's constant $a(G)$:     $1/7$

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