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Properties

Label 16T2
16T2 1 5 1->5 8 1->8 9 1->9 2 6 2->6 7 2->7 10 2->10 3 3->9 11 3->11 16 3->16 4 4->10 12 4->12 15 4->15 5->12 14 5->14 6->11 13 6->13 7->5 7->11 7->16 8->6 8->12 8->15 9->14 9->15 10->13 10->16 11->1 12->2 13->3 14->4 15->13 16->14
Degree $16$
Order $16$
Cyclic no
Abelian yes
Solvable yes
Primitive no
$p$-group yes
Group: $C_4\times C_2^2$

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Copy content magma:G := TransitiveGroup(16, 2);
 

Group invariants

Abstract group:  $C_4\times C_2^2$
Copy content magma:IdentifyGroup(G);
 
Order:  $16=2^{4}$
Copy content magma:Order(G);
 
Cyclic:  no
Copy content magma:IsCyclic(G);
 
Abelian:  yes
Copy content magma:IsAbelian(G);
 
Solvable:  yes
Copy content magma:IsSolvable(G);
 
Nilpotency class:  $1$
Copy content magma:NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
Copy content magma:t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $2$
Copy content magma:t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
Copy content magma:IsEven(G);
 
Primitive:  no
Copy content magma:IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $16$
Copy content magma:Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,9)(2,10)(3,11)(4,12)(5,14)(6,13)(7,16)(8,15)$, $(1,5)(2,6)(3,16)(4,15)(7,11)(8,12)(9,14)(10,13)$, $(1,8,6,11)(2,7,5,12)(3,9,15,13)(4,10,16,14)$
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
$8$:  $C_4\times C_2$ x 6, $C_2^3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_4$ x 4, $C_2^2$ x 7

Degree 8: $C_4\times C_2$ x 6, $C_2^3$

Low degree siblings

There are no siblings 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 $2^{8}$ $1$ $2$ $8$ $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,14)( 6,13)( 7,16)( 8,15)$
2B $2^{8}$ $1$ $2$ $8$ $( 1, 5)( 2, 6)( 3,16)( 4,15)( 7,11)( 8,12)( 9,14)(10,13)$
2C $2^{8}$ $1$ $2$ $8$ $( 1,13)( 2,14)( 3, 8)( 4, 7)( 5,10)( 6, 9)(11,15)(12,16)$
2D $2^{8}$ $1$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2E $2^{8}$ $1$ $2$ $8$ $( 1,14)( 2,13)( 3, 7)( 4, 8)( 5, 9)( 6,10)(11,16)(12,15)$
2F $2^{8}$ $1$ $2$ $8$ $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,13)( 6,14)( 7,15)( 8,16)$
2G $2^{8}$ $1$ $2$ $8$ $( 1, 6)( 2, 5)( 3,15)( 4,16)( 7,12)( 8,11)( 9,13)(10,14)$
4A1 $4^{4}$ $1$ $4$ $12$ $( 1, 8, 6,11)( 2, 7, 5,12)( 3, 9,15,13)( 4,10,16,14)$
4A-1 $4^{4}$ $1$ $4$ $12$ $( 1,11, 6, 8)( 2,12, 5, 7)( 3,13,15, 9)( 4,14,16,10)$
4B1 $4^{4}$ $1$ $4$ $12$ $( 1,15, 6, 3)( 2,16, 5, 4)( 7,14,12,10)( 8,13,11, 9)$
4B-1 $4^{4}$ $1$ $4$ $12$ $( 1, 3, 6,15)( 2, 4, 5,16)( 7,10,12,14)( 8, 9,11,13)$
4C1 $4^{4}$ $1$ $4$ $12$ $( 1,12, 6, 7)( 2,11, 5, 8)( 3,14,15,10)( 4,13,16, 9)$
4C-1 $4^{4}$ $1$ $4$ $12$ $( 1, 7, 6,12)( 2, 8, 5,11)( 3,10,15,14)( 4, 9,16,13)$
4D1 $4^{4}$ $1$ $4$ $12$ $( 1, 4, 6,16)( 2, 3, 5,15)( 7, 9,12,13)( 8,10,11,14)$
4D-1 $4^{4}$ $1$ $4$ $12$ $( 1,16, 6, 4)( 2,15, 5, 3)( 7,13,12, 9)( 8,14,11,10)$

Malle's constant $a(G)$:     $1/8$

Copy content magma:ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 4A1 4A-1 4B1 4B-1 4C1 4C-1 4D1 4D-1
Size 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
2 P 1A 1A 1A 1A 1A 1A 1A 1A 2G 2G 2G 2G 2G 2G 2G 2G
Type
16.10.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
16.10.1b R 1 −1 −1 1 1 −1 1 −1 −1 −1 −1 −1 1 1 1 1
16.10.1c R 1 −1 −1 1 1 −1 1 −1 1 1 1 1 −1 −1 −1 −1
16.10.1d R 1 −1 1 −1 1 −1 −1 1 −1 −1 1 1 −1 −1 1 1
16.10.1e R 1 −1 1 −1 1 −1 −1 1 1 1 −1 −1 1 1 −1 −1
16.10.1f R 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1 1 1 −1 −1
16.10.1g R 1 1 −1 −1 1 1 −1 −1 1 1 −1 −1 −1 −1 1 1
16.10.1h R 1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 −1 −1
16.10.1i1 C 1 −1 −1 1 −1 1 −1 1 −i i i −i i −i −i i
16.10.1i2 C 1 −1 −1 1 −1 1 −1 1 i −i −i i −i i i −i
16.10.1j1 C 1 −1 1 −1 −1 1 1 −1 −i i −i i −i i −i i
16.10.1j2 C 1 −1 1 −1 −1 1 1 −1 i −i i −i i −i i −i
16.10.1k1 C 1 1 −1 −1 −1 −1 1 1 −i i −i i i −i i −i
16.10.1k2 C 1 1 −1 −1 −1 −1 1 1 i −i i −i −i i −i i
16.10.1l1 C 1 1 1 1 −1 −1 −1 −1 −i i i −i −i i i −i
16.10.1l2 C 1 1 1 1 −1 −1 −1 −1 i −i −i i i −i −i i

Copy content magma:CharacterTable(G);
 

Regular extensions

Data not computed