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Properties

Label 16T681
16T681 1 13 1->13 14 1->14 2 2->13 2->14 3 15 3->15 3->15 4 16 4->16 4->16 5 10 5->10 5->10 6 9 6->9 6->9 7 11 7->11 12 7->12 8 8->11 8->12 9->1 9->3 10->2 10->4 11->6 11->8 12->5 12->7 13->6 13->7 14->5 14->8 15->2 16->1
Degree $16$
Order $256$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^2.C_2\wr C_4$

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Copy content magma:G := TransitiveGroup(16, 681);
 

Group invariants

Abstract group:  $C_2^2.C_2\wr C_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $256=2^{8}$
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 $t$:  $681$
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:  $(1,13,6,9)(2,14,5,10)(3,15)(4,16)(7,11,8,12)$, $(1,14,8,11,6,9,3,15,2,13,7,12,5,10,4,16)$
Copy content magma:Generators(G);
 

Low degree resolvents

$\card{(G/N)}$Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $C_2^2:C_4$
$32$:  $C_2^3 : C_4 $
$64$:  $((C_8 : C_2):C_2):C_2$
$128$:  16T397

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

16T680 x 2, 16T681, 32T3292, 32T3293 x 2, 32T3294, 32T3295, 32T3296, 32T7423, 32T7428

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 $2^{4},1^{8}$ $2$ $2$ $4$ $( 9,10)(11,12)(13,14)(15,16)$
2C $2^{8}$ $16$ $2$ $8$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,11)(10,12)(13,15)(14,16)$
2D $2^{6},1^{4}$ $16$ $2$ $6$ $( 3, 7)( 4, 8)( 5, 6)( 9,14)(10,13)(11,12)$
4A $4^{4}$ $4$ $4$ $12$ $( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,13,10,14)(11,15,12,16)$
4B $4^{2},2^{2},1^{4}$ $8$ $4$ $8$ $( 1, 5, 2, 6)( 3, 7, 4, 8)(11,12)(15,16)$
4C $4^{2},2^{4}$ $16$ $4$ $10$ $( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,11,10,12)(13,16,14,15)$
4D $4^{2},2^{2},1^{4}$ $16$ $4$ $8$ $( 3, 4)( 5, 6)( 9,16,10,15)(11,14,12,13)$
4E1 $4^{3},2^{2}$ $32$ $4$ $11$ $( 1,16)( 2,15)( 3,14, 7, 9)( 4,13, 8,10)( 5,12, 6,11)$
4E-1 $4^{3},2^{2}$ $32$ $4$ $11$ $( 1,16)( 2,15)( 3, 9, 7,14)( 4,10, 8,13)( 5,11, 6,12)$
8A1 $8^{2}$ $8$ $8$ $14$ $( 1, 7, 6, 4, 2, 8, 5, 3)( 9,16,13,11,10,15,14,12)$
8A-1 $8^{2}$ $8$ $8$ $14$ $( 1, 3, 5, 8, 2, 4, 6, 7)( 9,12,14,15,10,11,13,16)$
8B $8,4,1^{4}$ $16$ $8$ $10$ $( 1, 7, 5, 4, 2, 8, 6, 3)(11,16,12,15)$
8C $8,4,2^{2}$ $16$ $8$ $12$ $( 1, 2)( 3, 8, 4, 7)( 5, 6)( 9,12,14,16,10,11,13,15)$
16A1 $16$ $16$ $16$ $15$ $( 1,10, 7,15, 6,14, 4,12, 2, 9, 8,16, 5,13, 3,11)$
16A-1 $16$ $16$ $16$ $15$ $( 1,15, 8,14, 6,12, 3, 9, 2,16, 7,13, 5,11, 4,10)$
16B1 $16$ $16$ $16$ $15$ $( 1,13, 7,11, 6,10, 4,15, 2,14, 8,12, 5, 9, 3,16)$
16B-1 $16$ $16$ $16$ $15$ $( 1,11, 8,10, 6,15, 3,14, 2,12, 7, 9, 5,16, 4,13)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 4A 4B 4C 4D 4E1 4E-1 8A1 8A-1 8B 8C 16A1 16A-1 16B1 16B-1
Size 1 1 2 16 16 4 8 16 16 32 32 8 8 16 16 16 16 16 16
2 P 1A 1A 1A 1A 1A 2A 2B 2B 2B 2D 2D 4A 4A 4B 4B 8A1 8A-1 8A1 8A-1
Type
256.511.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
256.511.1b R 1 1 1 −1 1 1 1 1 −1 −1 −1 1 1 −1 −1 1 1 1 1
256.511.1c R 1 1 1 −1 1 1 1 1 −1 1 1 1 1 −1 −1 −1 −1 −1 −1
256.511.1d R 1 1 1 1 1 1 1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1
256.511.1e1 C 1 1 1 −1 −1 1 1 1 −1 −i i −1 −1 1 1 i i −i −i
256.511.1e2 C 1 1 1 −1 −1 1 1 1 −1 i −i −1 −1 1 1 −i −i i i
256.511.1f1 C 1 1 1 1 −1 1 1 1 1 −i i −1 −1 −1 −1 −i −i i i
256.511.1f2 C 1 1 1 1 −1 1 1 1 1 i −i −1 −1 −1 −1 i i −i −i
256.511.2a R 2 2 2 0 −2 2 2 −2 0 0 0 2 2 0 0 0 0 0 0
256.511.2b R 2 2 2 0 2 2 2 −2 0 0 0 −2 −2 0 0 0 0 0 0
256.511.4a R 4 4 4 0 0 4 −4 0 0 0 0 0 0 0 0 0 0 0 0
256.511.4b R 4 4 4 −2 0 −4 0 0 2 0 0 0 0 0 0 0 0 0 0
256.511.4c R 4 4 4 2 0 −4 0 0 −2 0 0 0 0 0 0 0 0 0 0
256.511.4d1 C 4 4 −4 0 0 0 0 0 0 0 0 −2i 2i 0 0 −1−i 1+i 1−i −1+i
256.511.4d2 C 4 4 −4 0 0 0 0 0 0 0 0 2i −2i 0 0 −1+i 1−i 1+i −1−i
256.511.4e1 C 4 4 −4 0 0 0 0 0 0 0 0 −2i 2i 0 0 1+i −1−i −1+i 1−i
256.511.4e2 C 4 4 −4 0 0 0 0 0 0 0 0 2i −2i 0 0 1−i −1+i −1−i 1+i
256.511.8a R 8 −8 0 0 0 0 0 0 0 0 0 0 0 −2 2 0 0 0 0
256.511.8b R 8 −8 0 0 0 0 0 0 0 0 0 0 0 2 −2 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed