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Properties

Label 16T1426
16T1426 1 7 1->7 1->7 11 1->11 2 8 2->8 2->8 12 2->12 3 5 3->5 3->5 9 3->9 4 6 4->6 4->6 10 4->10 14 5->14 16 5->16 13 6->13 15 6->15 7->2 7->14 7->16 8->1 8->13 8->15 9->4 9->13 10->3 10->14 11->1 11->2 11->15 12->1 12->2 12->16 13->8 13->11 13->11 14->7 14->12 14->12 15->5 15->10 15->10 16->6 16->9 16->9
Degree $16$
Order $2048$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^5.C_2\wr C_4$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(16, 1426);
 

Group invariants

Abstract group:  $C_2^5.C_2\wr C_4$
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:  $6$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1426$
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:  $(1,11,2,12)(3,9)(4,10)(5,14,7,16,6,13,8,15)$, $(1,7,2,8)(3,5)(4,6)(9,13,11,15,10,14,12,16)$, $(1,7,14,12,2,8,13,11)(3,5,16,9,4,6,15,10)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_4$ x 4, $C_2^2$ x 7
$8$:  $D_{4}$ x 8, $C_4\times C_2$ x 6, $C_2^3$
$16$:  $D_4\times C_2$ x 4, $C_2^2:C_4$ x 4, $Q_8:C_2$ x 2, $C_4\times C_2^2$
$32$:  $C_2^2 \wr C_2$, $C_2^3 : C_4 $ x 4, $C_4 \times D_4$ x 2, $C_2 \times (C_2^2:C_4)$, 16T34 x 2, 16T37
$64$:  $(C_4^2 : C_2):C_2$ x 2, $((C_8 : C_2):C_2):C_2$ x 2, $(((C_4 \times C_2): C_2):C_2):C_2$ x 2, 16T76 x 2, 32T239
$128$:  16T219, 16T222, 16T227, 16T230, 16T234, 16T235, 16T302
$256$:  32T3918, 32T4019, 32T4050
$512$:  16T940
$1024$:  32T63498

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $(((C_4 \times C_2): C_2):C_2):C_2$

Low degree siblings

16T1346 x 8, 16T1391 x 8, 16T1423 x 8, 16T1426 x 7, 32T97929 x 8, 32T97930 x 4, 32T97931 x 4, 32T97932 x 8, 32T97933 x 8, 32T97934 x 8, 32T97935 x 4, 32T97936 x 4, 32T97937 x 4, 32T97938 x 8, 32T97939 x 4, 32T98462 x 4, 32T98463 x 4, 32T98464 x 8, 32T98465 x 4, 32T98466 x 8, 32T98467 x 8, 32T98468 x 4, 32T98469 x 8, 32T98470 x 8, 32T98471 x 8, 32T98472 x 4, 32T98473 x 4, 32T98474 x 8, 32T98475 x 8, 32T98476 x 8, 32T98477 x 4, 32T98478 x 8, 32T98479 x 4, 32T98480 x 4, 32T98481 x 4, 32T98482 x 4, 32T98483 x 4, 32T98834 x 4, 32T98835 x 4, 32T98836 x 4, 32T98837 x 4, 32T98838 x 4, 32T98839 x 4, 32T98871 x 4, 32T98872 x 4, 32T98873 x 4, 32T98874 x 4, 32T98875 x 4, 32T98876 x 4, 32T115877 x 4, 32T141653 x 2, 32T142026 x 2, 32T142186 x 2, 32T142211 x 2, 32T142212 x 2, 32T142391 x 2, 32T175935 x 2, 32T180453 x 2, 32T180523 x 2, 32T182448 x 2, 32T182585 x 2, 32T182719 x 2, 32T191337 x 2, 32T204375 x 2

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

59 x 59 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed