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Properties

Label 16T45
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $D_8:C_2$

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Show commands: Magma

magma: G := TransitiveGroup(16, 45);
 

Group invariants

Abstract group:  $D_8:C_2$
magma: IdentifyGroup(G);
 
Order:  $32=2^{5}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $3$
magma: NilpotencyClass(G);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $45$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  $(1,9)(2,10)(3,12)(4,11)(5,14)(6,13)(7,16)(8,15)$, $(1,4,2,3)(5,16,6,15)(7,11,8,12)(9,14,10,13)$, $(1,11)(2,12)(3,9)(4,10)(5,8)(6,7)(13,15)(14,16)$
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 2, $C_2^3$
$16$:  $D_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4\times C_2$

Low degree siblings

8T15 x 2, 16T35, 16T38 x 2, 32T21

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

Malle's constant $a(G)$:     $1/6$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 4A 4B 4C 8A 8B
Size 1 1 2 4 4 4 2 2 4 4 4
2 P 1A 1A 1A 1A 1A 1A 2A 2A 2A 4A 4A
Type
32.43.1a R 1 1 1 1 1 1 1 1 1 1 1
32.43.1b R 1 1 −1 −1 1 −1 1 −1 1 1 −1
32.43.1c R 1 1 −1 −1 1 1 1 −1 −1 −1 1
32.43.1d R 1 1 −1 1 −1 −1 1 −1 1 −1 1
32.43.1e R 1 1 −1 1 −1 1 1 −1 −1 1 −1
32.43.1f R 1 1 1 −1 −1 −1 1 1 −1 1 1
32.43.1g R 1 1 1 −1 −1 1 1 1 1 −1 −1
32.43.1h R 1 1 1 1 1 −1 1 1 −1 −1 −1
32.43.2a R 2 2 −2 0 0 0 −2 2 0 0 0
32.43.2b R 2 2 2 0 0 0 −2 −2 0 0 0
32.43.4a R 4 −4 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);
 

Regular extensions

Data not computed