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Properties

Label 16T1668
16T1668 1 4 1->4 10 1->10 16 1->16 2 12 2->12 13 2->13 3 9 3->9 14 3->14 11 4->11 15 4->15 5 6 5->6 5->10 5->13 8 6->8 6->11 6->15 7 7->5 7->12 7->14 8->7 8->9 8->16 9->6 9->11 10->7 10->9 11->8 11->12 12->5 12->10 13->3 13->16 14->2 15->1 16->4
Degree $16$
Order $6144$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_2^6.(C_2^2\times S_4)$

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Show commands: Magma

Copy content magma:G := TransitiveGroup(16, 1668);
 

Group invariants

Abstract group:  $C_2^6.(C_2^2\times S_4)$
Copy content magma:IdentifyGroup(G);
 
Order:  $6144=2^{11} \cdot 3$
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:   not nilpotent
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1668$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $1$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,10)(2,12)(3,9)(4,11)(5,13)(6,15)(7,14)(8,16)$, $(1,16,4,15)(2,13,3,14)(5,10,7,12)(6,11,8,9)$, $(1,4)(5,6,8,7)(9,11,12,10)(13,16)$
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
$6$:  $S_3$
$8$:  $C_2^3$
$12$:  $D_{6}$ x 3
$24$:  $S_4$, $S_3 \times C_2^2$
$48$:  $S_4\times C_2$ x 3
$96$:  12T48
$192$:  $V_4^2:(S_3\times C_2)$ x 3
$384$:  12T136 x 3
$1536$:  24T4591

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 8: None

Low degree siblings

16T1668 x 7, 24T7847 x 8, 24T7848 x 8, 24T8961 x 24, 24T8965 x 24, 32T397141 x 12, 32T397333 x 4, 32T397439 x 8

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

65 x 65 character table

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed