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Properties

Label 16T41
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, 41);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $41$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $\OD_{16}:C_2$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $4$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (3,12)(4,11)(5,13)(6,14)(7,8)(15,16), (1,3,6,7,9,11,14,15)(2,4,5,8,10,12,13,16)
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: $C_2$

Degree 4: $C_4$, $D_{4}$ x 2

Degree 8: $C_2^2:C_4$, $(C_8:C_2):C_2$

Low degree siblings

8T16 x 2, 16T36, 16T41, 32T22

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

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);