↑

Properties

Label 16T165
16T165 1 2 1->2 3 1->3 14 1->14 4 2->4 13 2->13 3->2 3->4 11 3->11 4->1 12 4->12 5 8 5->8 9 5->9 6 7 6->7 10 6->10 7->5 15 7->15 8->6 16 8->16 9->4 9->11 9->16 10->3 10->12 10->15 11->6 11->10 11->13 12->5 12->9 12->14 13->8 13->12 13->16 14->7 14->11 14->15 15->1 15->9 15->13 16->2 16->10 16->14
Degree $16$
Order $64$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^3.D_4$

Related objects

Downloads

Learn more

Show commands: Magma

Copy content magma:G := TransitiveGroup(16, 165);
 

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:  $3$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $165$
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:  $(1,3,2,4)(5,8,6,7)(9,16,10,15)(11,13,12,14)$, $(1,14,7,15)(2,13,8,16)(3,11,6,10)(4,12,5,9)$, $(1,2)(3,4)(9,11,10,12)(13,16,14,15)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $D_{4}$ x 6, $C_2^3$
$16$:  $D_4\times C_2$ x 3
$32$:  $C_2^2 \wr C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4\times C_2$

Low degree siblings

16T146, 16T165 x 2, 16T173, 16T177 x 3, 32T162 x 3, 32T175 x 3

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

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C 4D1 4D-1 4E 4F 4G
Size 1 1 2 2 2 4 4 4 4 4 4 4 4 8 8 8
2 P 1A 1A 1A 1A 1A 1A 1A 1A 2A 2A 2A 2A 2A 2B 2C 2D
Type
64.139.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
64.139.1b R 1 1 1 1 1 −1 −1 1 −1 −1 1 −1 −1 −1 1 1
64.139.1c R 1 1 1 1 1 −1 −1 1 1 −1 1 1 −1 1 −1 −1
64.139.1d R 1 1 1 1 1 −1 1 −1 −1 −1 −1 −1 1 1 1 −1
64.139.1e R 1 1 1 1 1 −1 1 −1 1 −1 −1 1 1 −1 −1 1
64.139.1f R 1 1 1 1 1 1 −1 −1 −1 1 −1 −1 −1 1 −1 1
64.139.1g R 1 1 1 1 1 1 −1 −1 1 1 −1 1 −1 −1 1 −1
64.139.1h R 1 1 1 1 1 1 1 1 −1 1 1 −1 1 −1 −1 −1
64.139.2a R 2 2 −2 −2 2 −2 0 0 0 2 0 0 0 0 0 0
64.139.2b R 2 2 −2 −2 2 2 0 0 0 −2 0 0 0 0 0 0
64.139.2c R 2 2 −2 2 −2 0 0 −2 0 0 2 0 0 0 0 0
64.139.2d R 2 2 −2 2 −2 0 0 2 0 0 −2 0 0 0 0 0
64.139.2e R 2 2 2 −2 −2 0 −2 0 0 0 0 0 2 0 0 0
64.139.2f R 2 2 2 −2 −2 0 2 0 0 0 0 0 −2 0 0 0
64.139.4a1 C 4 −4 0 0 0 0 0 0 −2i 0 0 2i 0 0 0 0
64.139.4a2 C 4 −4 0 0 0 0 0 0 2i 0 0 −2i 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed