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Properties

Label 16T36
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $\OD_{16}:C_2$

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Show commands: Magma

magma: G := TransitiveGroup(16, 36);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $36$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\OD_{16}:C_2$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $8$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,3,6,8,10,12,13,15)(2,4,5,7,9,11,14,16), (1,2)(3,11)(4,12)(5,13)(6,14)(7,8)(9,10)(15,16)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $C_2^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $C_4\times C_2$, $(C_8:C_2):C_2$ x 2

Low degree siblings

8T16 x 2, 16T41 x 2, 32T22

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 3,12)( 4,11)( 7,16)( 8,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 2)( 3, 4)( 5,13)( 6,14)( 7,15)( 8,16)( 9,10)(11,12)$
$ 8, 8 $ $4$ $8$ $( 1, 3, 6, 8,10,12,13,15)( 2, 4, 5, 7, 9,11,14,16)$
$ 8, 8 $ $4$ $8$ $( 1, 4, 6,16,10,11,13, 7)( 2, 3, 5,15, 9,12,14, 8)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,13)(10,14)(11,15)(12,16)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 6,10,13)( 2, 5, 9,14)( 3, 8,12,15)( 4, 7,11,16)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 6,10,13)( 2, 5, 9,14)( 3,15,12, 8)( 4,16,11, 7)$
$ 8, 8 $ $4$ $8$ $( 1, 7,13,11,10,16, 6, 4)( 2, 8,14,12, 9,15, 5, 3)$
$ 8, 8 $ $4$ $8$ $( 1, 8, 6,12,10,15,13, 3)( 2, 7, 5,11, 9,16,14, 4)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)$

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:  $3$
Label:  32.7
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 4A 4B 8A1 8A-1 8B1 8B-1
Size 1 1 2 4 4 2 2 4 4 4 4
2 P 1A 1A 1A 1A 1A 2A 2A 4B 4A 4B 4A
Type
32.7.1a R 1 1 1 1 1 1 1 1 1 1 1
32.7.1b R 1 1 1 −1 −1 1 1 −1 −1 1 1
32.7.1c R 1 1 1 −1 −1 1 1 1 1 −1 −1
32.7.1d R 1 1 1 1 1 1 1 −1 −1 −1 −1
32.7.1e1 C 1 1 1 −1 1 −1 −1 −i i i −i
32.7.1e2 C 1 1 1 −1 1 −1 −1 i −i −i i
32.7.1f1 C 1 1 1 1 −1 −1 −1 −i i −i i
32.7.1f2 C 1 1 1 1 −1 −1 −1 i −i i −i
32.7.2a R 2 2 −2 0 0 −2 2 0 0 0 0
32.7.2b R 2 2 −2 0 0 2 −2 0 0 0 0
32.7.4a R 4 −4 0 0 0 0 0 0 0 0 0

magma: CharacterTable(G);