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Properties

Label 16T136
16T136 1 5 1->5 15 1->15 2 6 2->6 16 2->16 3 4 3->4 13 3->13 14 4->14 12 5->12 11 6->11 7 10 7->10 8 9 8->9 9->4 9->16 10->3 10->15 11->1 11->14 12->2 12->13 13->8 13->11 14->7 14->12 15->5 15->9 16->6 16->10
Degree $16$
Order $64$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_{16}:C_4$

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Copy content magma:G := TransitiveGroup(16, 136);
 

Group invariants

Abstract group:  $C_{16}:C_4$
Copy content magma:IdentifyGroup(G);
 
Order:  $64=2^{6}$
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 $t$:  $136$
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:  $(1,5)(2,6)(3,4)(9,16,10,15)(11,14,12,13)$, $(1,15,5,12,2,16,6,11)(3,13,8,9,4,14,7,10)$
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, $C_2^2$
$8$:  $D_{4}$, $C_4\times C_2$, $Q_8$
$16$:  $QD_{16}$ x 2, $C_4:C_4$
$32$:  32T47

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $QD_{16}$

Low degree siblings

32T149

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 $2^{4},1^{8}$ $2$ $2$ $4$ $( 9,10)(11,12)(13,14)(15,16)$
4A $4^{4}$ $2$ $4$ $12$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,13,10,14)(11,16,12,15)$
4B $4^{4}$ $2$ $4$ $12$ $( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,13,10,14)(11,16,12,15)$
4C1 $4^{2},2^{3},1^{2}$ $8$ $4$ $9$ $( 3, 8)( 4, 7)( 5, 6)( 9,11,10,12)(13,15,14,16)$
4C-1 $4^{2},2^{3},1^{2}$ $8$ $4$ $9$ $( 3, 8)( 4, 7)( 5, 6)( 9,12,10,11)(13,16,14,15)$
8A1 $8^{2}$ $2$ $8$ $14$ $( 1, 3, 5, 8, 2, 4, 6, 7)( 9,11,13,16,10,12,14,15)$
8A-1 $8^{2}$ $2$ $8$ $14$ $( 1, 7, 6, 4, 2, 8, 5, 3)( 9,15,14,12,10,16,13,11)$
8B $8^{2}$ $4$ $8$ $14$ $( 1, 3, 5, 8, 2, 4, 6, 7)( 9,12,13,15,10,11,14,16)$
8C1 $8^{2}$ $8$ $8$ $14$ $( 1,13, 6,10, 2,14, 5, 9)( 3,12, 7,15, 4,11, 8,16)$
8C-1 $8^{2}$ $8$ $8$ $14$ $( 1,13, 5, 9, 2,14, 6,10)( 3,12, 8,16, 4,11, 7,15)$
16A1 $16$ $4$ $16$ $15$ $( 1,13, 3,16, 5,10, 8,12, 2,14, 4,15, 6, 9, 7,11)$
16A-1 $16$ $4$ $16$ $15$ $( 1,10, 4,11, 5,14, 7,16, 2, 9, 3,12, 6,13, 8,15)$
16B1 $16$ $4$ $16$ $15$ $( 1,13, 4,15, 5,10, 7,11, 2,14, 3,16, 6, 9, 8,12)$
16B-1 $16$ $4$ $16$ $15$ $( 1,10, 3,12, 5,14, 8,15, 2, 9, 4,11, 6,13, 7,16)$

Malle's constant $a(G)$:     $1/4$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 4A 4B 4C1 4C-1 8A1 8A-1 8B 8C1 8C-1 16A1 16A-1 16B1 16B-1
Size 1 1 2 2 2 8 8 2 2 4 8 8 4 4 4 4
2 P 1A 1A 1A 2A 2A 2B 2B 4A 4A 4A 4B 4B 8A1 8A-1 8A-1 8A1
Type
64.46.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
64.46.1b R 1 1 1 1 1 −1 −1 1 1 1 −1 −1 1 1 1 1
64.46.1c R 1 1 1 1 1 −1 −1 1 1 1 1 1 −1 −1 −1 −1
64.46.1d R 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1
64.46.1e1 C 1 1 −1 1 −1 −i i 1 1 −1 −i i −1 1 −1 1
64.46.1e2 C 1 1 −1 1 −1 i −i 1 1 −1 i −i −1 1 −1 1
64.46.1f1 C 1 1 −1 1 −1 −i i 1 1 −1 i −i 1 −1 1 −1
64.46.1f2 C 1 1 −1 1 −1 i −i 1 1 −1 −i i 1 −1 1 −1
64.46.2a R 2 2 2 2 2 0 0 −2 −2 −2 0 0 0 0 0 0
64.46.2b S 2 2 −2 2 −2 0 0 −2 −2 2 0 0 0 0 0 0
64.46.2c1 C 2 2 −2 −2 2 0 0 0 0 0 0 0 −ζ8−ζ83 −ζ8−ζ83 ζ8+ζ83 ζ8+ζ83
64.46.2c2 C 2 2 −2 −2 2 0 0 0 0 0 0 0 ζ8+ζ83 ζ8+ζ83 −ζ8−ζ83 −ζ8−ζ83
64.46.2d1 C 2 2 2 −2 −2 0 0 0 0 0 0 0 −ζ8−ζ83 ζ8+ζ83 ζ8+ζ83 −ζ8−ζ83
64.46.2d2 C 2 2 2 −2 −2 0 0 0 0 0 0 0 ζ8+ζ83 −ζ8−ζ83 −ζ8−ζ83 ζ8+ζ83
64.46.4a1 C 4 −4 0 0 0 0 0 −2ζ8−2ζ83 2ζ8+2ζ83 0 0 0 0 0 0 0
64.46.4a2 C 4 −4 0 0 0 0 0 2ζ8+2ζ83 −2ζ8−2ζ83 0 0 0 0 0 0 0

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed