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Properties

Label 16T160
16T160 1 3 1->3 5 1->5 2 4 2->4 6 2->6 3->2 7 3->7 4->1 8 4->8 5->2 13 5->13 6->1 14 6->14 7->4 16 7->16 8->3 15 8->15 9 11 9->11 9->15 10 12 10->12 10->16 11->10 11->14 12->9 12->13
Degree $16$
Order $64$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^3.D_4$

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Copy content magma:G := TransitiveGroup(16, 160);
 

Group invariants

Abstract group:  $C_2^3.D_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $64=2^{6}$
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 $t$:  $160$
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:  $(1,5,2,6)(3,7,4,8)(9,15)(10,16)(11,14)(12,13)$, $(1,3,2,4)(5,13)(6,14)(7,16)(8,15)(9,11,10,12)$
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, $C_4\times C_2$
$16$:  $C_2^2:C_4$
$32$:  $C_2^3 : C_4 $

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4$

Low degree siblings

16T140, 16T148, 16T153, 32T153, 32T154, 32T164

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 $2^{4},1^{8}$ $2$ $2$ $4$ $( 1, 2)( 3, 4)( 9,10)(11,12)$
2C $2^{8}$ $4$ $2$ $8$ $( 1,10)( 2, 9)( 3,11)( 4,12)( 5,15)( 6,16)( 7,13)( 8,14)$
2D $2^{4},1^{8}$ $4$ $2$ $4$ $(1,2)(3,4)(5,6)(7,8)$
4A $4^{4}$ $4$ $4$ $12$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,15, 6,16)( 7,13, 8,14)$
4B1 $4^{4}$ $4$ $4$ $12$ $( 1, 4, 2, 3)( 5,14, 6,13)( 7,15, 8,16)( 9,11,10,12)$
4B-1 $4^{4}$ $4$ $4$ $12$ $( 1, 3, 2, 4)( 5,14, 6,13)( 7,15, 8,16)( 9,12,10,11)$
4C $4^{2},2^{4}$ $8$ $4$ $10$ $( 1, 3, 2, 4)( 5,13)( 6,14)( 7,16)( 8,15)( 9,11,10,12)$
4D1 $4^{2},2^{4}$ $8$ $4$ $10$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,15)(10,16)(11,14)(12,13)$
4D-1 $4^{2},2^{4}$ $8$ $4$ $10$ $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,15)(10,16)(11,14)(12,13)$
8A1 $8^{2}$ $8$ $8$ $14$ $( 1,13, 9, 8, 2,14,10, 7)( 3,16,12, 5, 4,15,11, 6)$
8A-1 $8^{2}$ $8$ $8$ $14$ $( 1,14,10, 7, 2,13, 9, 8)( 3,15,11, 6, 4,16,12, 5)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 4A 4B1 4B-1 4C 4D1 4D-1 8A1 8A-1
Size 1 1 2 4 4 4 4 4 8 8 8 8 8
2 P 1A 1A 1A 1A 1A 2A 2A 2A 2B 2D 2D 4A 4A
Type
64.33.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1
64.33.1b R 1 1 1 1 1 1 −1 −1 −1 −1 −1 1 1
64.33.1c R 1 1 1 1 1 1 −1 −1 −1 1 1 −1 −1
64.33.1d R 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
64.33.1e1 C 1 1 1 1 −1 −1 −1 −1 1 −i i −i i
64.33.1e2 C 1 1 1 1 −1 −1 −1 −1 1 i −i i −i
64.33.1f1 C 1 1 1 1 −1 −1 1 1 −1 −i i i −i
64.33.1f2 C 1 1 1 1 −1 −1 1 1 −1 i −i −i i
64.33.2a R 2 2 2 −2 −2 2 0 0 0 0 0 0 0
64.33.2b R 2 2 2 −2 2 −2 0 0 0 0 0 0 0
64.33.4a R 4 4 −4 0 0 0 0 0 0 0 0 0 0
64.33.4b1 C 4 −4 0 0 0 0 −2i 2i 0 0 0 0 0
64.33.4b2 C 4 −4 0 0 0 0 2i −2i 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed