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Properties

Label 16T877
16T877 1 4 1->4 5 1->5 12 1->12 1->12 2 3 2->3 6 2->6 11 2->11 2->11 7 3->7 9 3->9 10 3->10 8 4->8 4->9 4->10 5->7 15 5->15 6->8 16 6->16 14 7->14 13 8->13 9->3 9->4 9->10 9->12 9->16 10->3 10->4 10->11 10->15 11->1 11->1 11->12 11->13 12->2 12->2 12->14 13->14 13->15 14->16 15->16
Degree $16$
Order $512$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $D_4^2:D_4$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(16, 877);
 

Group invariants

Abstract group:  $D_4^2:D_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $512=2^{9}$
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 $t$:  $877$
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:  $(1,12,2,11)(3,10,4,9)(5,15)(6,16)(7,14)(8,13)$, $(1,5)(2,6)(3,7)(4,8)(9,16)(10,15)(11,13)(12,14)$, $(9,10)(11,12)(13,14)(15,16)$, $(1,4)(2,3)(5,7)(6,8)(9,12)(10,11)(13,15)(14,16)$, $(1,12,2,11)(3,9,4,10)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 15
$4$:  $C_2^2$ x 35
$8$:  $D_{4}$ x 28, $C_2^3$ x 15
$16$:  $D_4\times C_2$ x 42, $C_2^4$
$32$:  $C_2^2 \wr C_2$ x 28, $C_2^2 \times D_4$ x 7
$64$:  16T105 x 7
$128$:  16T223, 16T239, 16T325
$256$:  32T3930

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$ x 3

Degree 8: $C_2^2 \wr C_2$

Low degree siblings

16T837 x 8, 16T877 x 7, 16T935 x 8, 32T10047 x 8, 32T10048 x 4, 32T10049 x 4, 32T10050 x 8, 32T10051 x 4, 32T10052 x 4, 32T10053 x 4, 32T10054 x 4, 32T10055 x 4, 32T10056 x 4, 32T10057 x 4, 32T10058 x 4, 32T10059 x 4, 32T10060 x 4, 32T10061 x 4, 32T10326 x 8, 32T10327 x 4, 32T10328 x 4, 32T10329 x 4, 32T10330 x 4, 32T10331 x 4, 32T10332 x 4, 32T10333 x 4, 32T10334 x 4, 32T10335 x 4, 32T10336 x 4, 32T10337 x 4, 32T10338 x 4, 32T10339 x 4, 32T10646 x 4, 32T10647 x 4, 32T10648 x 4, 32T10649 x 4, 32T10650 x 4, 32T10651 x 4, 32T10652 x 4, 32T10653 x 4, 32T10654 x 4, 32T10655 x 4, 32T10656 x 4, 32T10657 x 4, 32T10658 x 4, 32T22005 x 4, 32T22085 x 4

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

53 x 53 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed