↑

Properties

Label 16T26
16T26 1 3 1->3 5 1->5 2 4 2->4 6 2->6 3->2 3->4 4->1 16 5->16 15 6->15 7 13 7->13 7->15 8 14 8->14 8->16 9 11 9->11 9->14 10 12 10->12 10->13 11->10 11->12 12->9 13->8 14->7 15->5 16->6
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $D_4:C_4$

Related objects

Downloads

Learn more

Show commands: Magma

Copy content magma:G := TransitiveGroup(16, 26);
 

Group invariants

Abstract group:  $D_4:C_4$
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:  $3$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $26$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $4$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,3,2,4)(5,16,6,15)(7,13,8,14)(9,11,10,12)$, $(1,5)(2,6)(3,4)(7,15)(8,16)(9,14)(10,13)(11,12)$
Copy content magma:Generators(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, $C_4\times C_2$
$16$:  $D_{8}$, $QD_{16}$, $C_2^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

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

Degree 8: $D_{8}$, $QD_{16}$, $C_2^2:C_4$

Low degree siblings

16T26, 32T12

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 $2^{8}$ $1$ $2$ $8$ $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,13)( 6,14)( 7,15)( 8,16)$
2B $2^{8}$ $1$ $2$ $8$ $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,14)( 6,13)( 7,16)( 8,15)$
2C $2^{8}$ $1$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2D $2^{6},1^{4}$ $4$ $2$ $6$ $( 3,15)( 4,16)( 5,14)( 6,13)( 7,12)( 8,11)$
2E $2^{8}$ $4$ $2$ $8$ $( 1,14)( 2,13)( 3,12)( 4,11)( 5, 9)( 6,10)( 7, 8)(15,16)$
4A $4^{4}$ $2$ $4$ $12$ $( 1, 6, 9,13)( 2, 5,10,14)( 3, 8,11,15)( 4, 7,12,16)$
4B $4^{4}$ $2$ $4$ $12$ $( 1,14, 9, 5)( 2,13,10, 6)( 3,16,11, 7)( 4,15,12, 8)$
4C1 $4^{4}$ $4$ $4$ $12$ $( 1,15, 2,16)( 3,14, 4,13)( 5,12, 6,11)( 7, 9, 8,10)$
4C-1 $4^{4}$ $4$ $4$ $12$ $( 1,12, 2,11)( 3, 9, 4,10)( 5, 8, 6, 7)(13,16,14,15)$
8A1 $8^{2}$ $2$ $8$ $14$ $( 1,15,14,12, 9, 8, 5, 4)( 2,16,13,11,10, 7, 6, 3)$
8A-1 $8^{2}$ $2$ $8$ $14$ $( 1, 4, 5, 8, 9,12,14,15)( 2, 3, 6, 7,10,11,13,16)$
8A3 $8^{2}$ $2$ $8$ $14$ $( 1,12, 5,15, 9, 4,14, 8)( 2,11, 6,16,10, 3,13, 7)$
8A-3 $8^{2}$ $2$ $8$ $14$ $( 1, 8,14, 4, 9,15, 5,12)( 2, 7,13, 3,10,16, 6,11)$

Malle's constant $a(G)$:     $1/6$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 4A 4B 4C1 4C-1 8A1 8A-1 8A3 8A-3
Size 1 1 1 1 4 4 2 2 4 4 2 2 2 2
2 P 1A 1A 1A 1A 1A 1A 2B 2B 2C 2C 4B 4B 4B 4B
Type
32.9.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.9.1b R 1 1 1 1 −1 −1 1 1 −1 −1 1 1 1 1
32.9.1c R 1 1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1
32.9.1d R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
32.9.1e1 C 1 −1 1 −1 −1 1 1 −1 −i i i i −i −i
32.9.1e2 C 1 −1 1 −1 −1 1 1 −1 i −i −i −i i i
32.9.1f1 C 1 −1 1 −1 1 −1 1 −1 −i i −i −i i i
32.9.1f2 C 1 −1 1 −1 1 −1 1 −1 i −i i i −i −i
32.9.2a R 2 −2 2 −2 0 0 −2 2 0 0 0 0 0 0
32.9.2b R 2 2 2 2 0 0 −2 −2 0 0 0 0 0 0
32.9.2c1 R 2 −2 −2 2 0 0 0 0 0 0 −ζ8−1−ζ8 ζ8−1+ζ8 −ζ8−1−ζ8 ζ8−1+ζ8
32.9.2c2 R 2 −2 −2 2 0 0 0 0 0 0 ζ8−1+ζ8 −ζ8−1−ζ8 ζ8−1+ζ8 −ζ8−1−ζ8
32.9.2d1 C 2 2 −2 −2 0 0 0 0 0 0 −ζ8−ζ83 ζ8+ζ83 ζ8+ζ83 −ζ8−ζ83
32.9.2d2 C 2 2 −2 −2 0 0 0 0 0 0 ζ8+ζ83 −ζ8−ζ83 −ζ8−ζ83 ζ8+ζ83

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed