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Properties

Label 16T1461
16T1461 1 4 1->4 6 1->6 7 1->7 2 3 2->3 5 2->5 8 2->8 3->6 14 3->14 15 3->15 4->5 13 4->13 16 4->16 11 5->11 5->16 12 6->12 6->15 10 7->10 7->12 9 8->9 8->11 9->11 9->13 9->16 10->12 10->14 10->15 11->6 11->7 11->13 12->5 12->8 12->14 13->3 13->8 13->10 14->4 14->7 14->9 15->2 15->4 16->1 16->3
Degree $16$
Order $2048$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2\wr D_4$

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Copy content magma:G := TransitiveGroup(16, 1461);
 

Group invariants

Abstract group:  $C_2\wr D_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $2048=2^{11}$
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:  $5$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1461$
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:  $(1,6)(2,5)(3,15,4,16)(7,12,8,11)(9,13,10,14)$, $(1,7)(2,8)(3,14,4,13)(5,11,6,12)(9,16)(10,15)$, $(1,4,5,16)(2,3,6,15)(7,10,12,14)(8,9,11,13)$
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 12, $C_2^3$
$16$:  $D_4\times C_2$ x 6, $Q_8:C_2$
$32$:  $C_2^2 \wr C_2$ x 3, 16T34 x 3, $C_4^2:C_2$
$64$:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 6, 32T320
$128$:  $C_2 \wr C_2\wr C_2$ x 4, 16T342 x 4, 16T350 x 3
$256$:  32T5807 x 4, 32T6030
$512$:  16T956, 16T969 x 2
$1024$:  32T41928

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

16T1439 x 16, 16T1445 x 64, 16T1461 x 15, 32T99003 x 16, 32T99004 x 16, 32T99005 x 8, 32T99006 x 16, 32T99007 x 8, 32T99008 x 8, 32T99009 x 16, 32T99010 x 8, 32T99011 x 16, 32T99012 x 8, 32T99013 x 8, 32T99053 x 64, 32T99054 x 32, 32T99055 x 128, 32T99056 x 32, 32T99057 x 32, 32T99058 x 128, 32T99059 x 32, 32T99060 x 32, 32T99061 x 32, 32T99197 x 64, 32T99198 x 32, 32T99199 x 128, 32T99200 x 128, 32T99201 x 32, 32T99202 x 16, 32T99203 x 32, 32T99204 x 16, 32T99205 x 128, 32T99206 x 64, 32T99207 x 32, 32T99208 x 32, 32T99209 x 16, 32T99210 x 8, 32T99211 x 16, 32T99212 x 8, 32T99213 x 16, 32T99214 x 8, 32T99215 x 16, 32T99216 x 8, 32T99217 x 32, 32T99218 x 32, 32T99219 x 32, 32T99220 x 8, 32T99221 x 8, 32T99222 x 32, 32T99223 x 16, 32T99224 x 16, 32T110223 x 8, 32T116047 x 16

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

Conjugacy classes not computed

Copy content magma:ConjugacyClasses(G);
 

Character table

74 x 74 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed