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Properties

Label 16T34
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_4:D_4$

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Show commands: Magma

magma: G := TransitiveGroup(16, 34);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $34$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_4:D_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,11,16,13)(2,12,15,14)(3,9,5,7)(4,10,6,8), (1,15)(2,16)(3,5)(4,6)(7,8)(9,10), (1,10,16,8)(2,9,15,7)(3,12,5,14)(4,11,6,13)
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
$8$:  $D_{4}$ x 4, $C_2^3$
$16$:  $D_4\times C_2$ x 2, $Q_8:C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 4

Degree 8: $D_4\times C_2$ x 2, $Q_8:C_2$

Low degree siblings

16T34, 16T43 x 2, 32T20

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

Malle's constant $a(G)$:     $1/6$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $32=2^{5}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $2$
Label:  32.28
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 4A 4B 4C1 4C-1 4D 4E
Size 1 1 1 1 2 2 4 4 2 2 2 2 4 4
2 P 1A 1A 1A 1A 1A 1A 1A 1A 2B 2B 2B 2B 2C 2C
Type
32.28.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.28.1b R 1 1 1 1 −1 −1 −1 −1 1 1 −1 −1 1 1
32.28.1c R 1 1 1 1 −1 −1 −1 1 −1 −1 1 1 −1 1
32.28.1d R 1 1 1 1 −1 −1 1 −1 −1 −1 1 1 1 −1
32.28.1e R 1 1 1 1 −1 −1 1 1 1 1 −1 −1 −1 −1
32.28.1f R 1 1 1 1 1 1 −1 −1 1 1 1 1 −1 −1
32.28.1g R 1 1 1 1 1 1 −1 1 −1 −1 −1 −1 1 −1
32.28.1h R 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 −1 1
32.28.2a R 2 −2 2 −2 −2 2 0 0 0 0 0 0 0 0
32.28.2b R 2 −2 2 −2 2 −2 0 0 0 0 0 0 0 0
32.28.2c R 2 2 −2 −2 0 0 0 0 −2 2 0 0 0 0
32.28.2d R 2 2 −2 −2 0 0 0 0 2 −2 0 0 0 0
32.28.2e1 C 2 −2 −2 2 0 0 0 0 0 0 −2i 2i 0 0
32.28.2e2 C 2 −2 −2 2 0 0 0 0 0 0 2i −2i 0 0

magma: CharacterTable(G);