↑

Properties

Label 16T29
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2\times D_8$

Related objects

Downloads

Learn more

Show commands: Magma

magma: G := TransitiveGroup(16, 29);
 

Group invariants

Abstract group:  $C_2\times D_8$
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 $t$:  $29$
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:  $(1,9)(2,10)(3,12)(4,11)(5,13)(6,14)(7,15)(8,16)$, $(1,12,5,16,10,4,14,7)(2,11,6,15,9,3,13,8)$, $(1,12)(2,11)(3,9)(4,10)(5,7)(6,8)(13,15)(14,16)$
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, $C_2^3$
$16$:  $D_{8}$ x 2, $D_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_{8}$ x 2, $D_4\times C_2$

Low degree siblings

16T29 x 3, 32T15

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 $2^{8}$ $1$ $2$ $8$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,13)( 6,14)( 7,15)( 8,16)$
2B $2^{8}$ $1$ $2$ $8$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
2C $2^{8}$ $1$ $2$ $8$ $( 1,10)( 2, 9)( 3,11)( 4,12)( 5,14)( 6,13)( 7,16)( 8,15)$
2D $2^{8}$ $4$ $2$ $8$ $( 1, 3)( 2, 4)( 5,15)( 6,16)( 7,13)( 8,14)( 9,12)(10,11)$
2E $2^{8}$ $4$ $2$ $8$ $( 1,12)( 2,11)( 3, 9)( 4,10)( 5, 7)( 6, 8)(13,15)(14,16)$
2F $2^{6},1^{4}$ $4$ $2$ $6$ $( 3,15)( 4,16)( 5,14)( 6,13)( 7,12)( 8,11)$
2G $2^{8}$ $4$ $2$ $8$ $( 1, 9)( 2,10)( 3, 7)( 4, 8)( 5, 6)(11,16)(12,15)(13,14)$
4A $4^{4}$ $2$ $4$ $12$ $( 1,13,10, 6)( 2,14, 9, 5)( 3,16,11, 7)( 4,15,12, 8)$
4B $4^{4}$ $2$ $4$ $12$ $( 1, 5,10,14)( 2, 6, 9,13)( 3, 8,11,15)( 4, 7,12,16)$
8A1 $8^{2}$ $2$ $8$ $14$ $( 1, 4, 5, 7,10,12,14,16)( 2, 3, 6, 8, 9,11,13,15)$
8A3 $8^{2}$ $2$ $8$ $14$ $( 1,12, 5,16,10, 4,14, 7)( 2,11, 6,15, 9, 3,13, 8)$
8B1 $8^{2}$ $2$ $8$ $14$ $( 1,11, 5,15,10, 3,14, 8)( 2,12, 6,16, 9, 4,13, 7)$
8B3 $8^{2}$ $2$ $8$ $14$ $( 1, 3, 5, 8,10,11,14,15)( 2, 4, 6, 7, 9,12,13,16)$

Malle's constant $a(G)$:     $1/6$

magma: ConjugacyClasses(G);
 

Character table

1A 2A 2B 2C 2D 2E 2F 2G 4A 4B 8A1 8A3 8B1 8B3
Size 1 1 1 1 4 4 4 4 2 2 2 2 2 2
2 P 1A 1A 1A 1A 1A 1A 1A 1A 2C 2C 4B 4B 4B 4B
Type
32.39.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.39.1b R 1 1 −1 −1 −1 −1 1 1 1 −1 −1 −1 1 1
32.39.1c R 1 1 −1 −1 −1 1 −1 1 1 −1 1 1 −1 −1
32.39.1d R 1 1 −1 −1 1 −1 1 −1 1 −1 1 1 −1 −1
32.39.1e R 1 1 −1 −1 1 1 −1 −1 1 −1 −1 −1 1 1
32.39.1f R 1 1 1 1 −1 −1 −1 −1 1 1 1 1 1 1
32.39.1g R 1 1 1 1 −1 1 1 −1 1 1 −1 −1 −1 −1
32.39.1h R 1 1 1 1 1 −1 −1 1 1 1 −1 −1 −1 −1
32.39.2a R 2 2 −2 −2 0 0 0 0 −2 2 0 0 0 0
32.39.2b R 2 2 2 2 0 0 0 0 −2 −2 0 0 0 0
32.39.2c1 R 2 −2 −2 2 0 0 0 0 0 0 −ζ8−1−ζ8 ζ8−1+ζ8 −ζ8−1−ζ8 ζ8−1+ζ8
32.39.2c2 R 2 −2 −2 2 0 0 0 0 0 0 ζ8−1+ζ8 −ζ8−1−ζ8 ζ8−1+ζ8 −ζ8−1−ζ8
32.39.2d1 R 2 −2 2 −2 0 0 0 0 0 0 −ζ8−1−ζ8 ζ8−1+ζ8 ζ8−1+ζ8 −ζ8−1−ζ8
32.39.2d2 R 2 −2 2 −2 0 0 0 0 0 0 ζ8−1+ζ8 −ζ8−1−ζ8 −ζ8−1−ζ8 ζ8−1+ζ8

magma: CharacterTable(G);
 

Regular extensions

Data not computed