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Properties

Label 16T28
16T28 1 7 1->7 13 1->13 2 8 2->8 14 2->14 3 10 3->10 15 3->15 4 9 4->9 16 4->16 5 5->10 11 5->11 6 6->9 12 6->12 7->6 7->11 8->5 8->12 9->5 9->16 10->6 10->15 11->1 11->8 12->2 12->7 13->2 13->4 14->1 14->3 15->4 15->13 16->3 16->14
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_4\wr C_2$

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Copy content magma:G := TransitiveGroup(16, 28);
 

Group invariants

Abstract group:  $C_4\wr C_2$
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 $t$:  $28$
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:  $(1,13,2,14)(3,15,4,16)(5,10,6,9)(7,11,8,12)$, $(1,7,6,12,2,8,5,11)(3,10,15,13,4,9,16,14)$
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, $C_4\times C_2$
$16$:  $C_2^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

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

Degree 8: $C_2^2:C_4$

Low degree siblings

8T17 x 2, 16T42, 32T14

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 $2^{8}$ $2$ $2$ $8$ $( 1,13)( 2,14)( 3, 7)( 4, 8)( 5,10)( 6, 9)(11,16)(12,15)$
2C $2^{6},1^{4}$ $4$ $2$ $6$ $( 1, 2)( 3,11)( 4,12)( 5, 6)( 7,15)( 8,16)$
4A1 $4^{4}$ $1$ $4$ $12$ $( 1, 6, 2, 5)( 3,15, 4,16)( 7,12, 8,11)( 9,14,10,13)$
4A-1 $4^{4}$ $1$ $4$ $12$ $( 1, 5, 2, 6)( 3,16, 4,15)( 7,11, 8,12)( 9,13,10,14)$
4B $4^{4}$ $2$ $4$ $12$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,13, 6,14)( 7,15, 8,16)$
4C1 $4^{4}$ $2$ $4$ $12$ $( 1,16,13,11)( 2,15,14,12)( 3, 9, 7, 6)( 4,10, 8, 5)$
4C-1 $4^{4}$ $2$ $4$ $12$ $( 1,11,13,16)( 2,12,14,15)( 3, 6, 7, 9)( 4, 5, 8,10)$
4D1 $4^{4}$ $2$ $4$ $12$ $( 1, 3,14, 8)( 2, 4,13, 7)( 5,16, 9,12)( 6,15,10,11)$
4D-1 $4^{4}$ $2$ $4$ $12$ $( 1, 8,14, 3)( 2, 7,13, 4)( 5,12, 9,16)( 6,11,10,15)$
4E $4^{4}$ $4$ $4$ $12$ $( 1,13, 2,14)( 3,15, 4,16)( 5,10, 6, 9)( 7,11, 8,12)$
8A1 $8^{2}$ $4$ $8$ $14$ $( 1, 8, 6,11, 2, 7, 5,12)( 3, 9,15,14, 4,10,16,13)$
8A-1 $8^{2}$ $4$ $8$ $14$ $( 1, 3, 5,16, 2, 4, 6,15)( 7, 9,11,13, 8,10,12,14)$

Malle's constant $a(G)$:     $1/6$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 4A1 4A-1 4B 4C1 4C-1 4D1 4D-1 4E 8A1 8A-1
Size 1 1 2 4 1 1 2 2 2 2 2 4 4 4
2 P 1A 1A 1A 1A 2A 2A 2A 2B 2B 2B 2B 2A 4A1 4A-1
Type
32.11.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.11.1b R 1 1 1 −1 1 1 −1 −1 −1 1 −1 −1 1 1
32.11.1c R 1 1 1 −1 1 1 1 1 1 1 1 −1 −1 −1
32.11.1d R 1 1 1 1 1 1 −1 −1 −1 1 −1 1 −1 −1
32.11.1e1 C 1 1 −1 −1 −1 −1 −i −i i 1 i 1 i −i
32.11.1e2 C 1 1 −1 −1 −1 −1 i i −i 1 −i 1 −i i
32.11.1f1 C 1 1 −1 1 −1 −1 −i −i i 1 i −1 −i i
32.11.1f2 C 1 1 −1 1 −1 −1 i i −i 1 −i −1 i −i
32.11.2a R 2 2 −2 0 2 2 0 0 0 −2 0 0 0 0
32.11.2b R 2 2 2 0 −2 −2 0 0 0 −2 0 0 0 0
32.11.2c1 C 2 −2 0 0 −2i 2i −1+i 1−i 1+i 0 −1−i 0 0 0
32.11.2c2 C 2 −2 0 0 2i −2i −1−i 1+i 1−i 0 −1+i 0 0 0
32.11.2d1 C 2 −2 0 0 −2i 2i 1−i −1+i −1−i 0 1+i 0 0 0
32.11.2d2 C 2 −2 0 0 2i −2i 1+i −1−i −1+i 0 1−i 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed