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Properties

Label 16T39
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^2\wr C_2$

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Show commands: Magma

magma: G := TransitiveGroup(16, 39);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $39$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^2\wr C_2$
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:  (1,9,16,8)(2,10,15,7)(3,11,6,14)(4,12,5,13), (1,7,5,11)(2,8,6,12)(3,13,15,9)(4,14,16,10), (1,2)(3,4)(5,6)(7,9)(8,10)(11,13)(12,14)(15,16)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 7
$4$:  $C_2^2$ x 7
$8$:  $D_{4}$ x 6, $C_2^3$
$16$:  $D_4\times C_2$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 6

Degree 8: $D_4$, $D_4\times C_2$ x 2, $C_2^2 \wr C_2$ x 4

Low degree siblings

8T18 x 8, 16T39 x 5, 16T46, 32T24

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

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:  32.27
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 2F 2G 2H 2I 2J 4A 4B 4C
Size 1 1 1 1 2 2 2 2 2 2 4 4 4 4
2 P 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 1A 2A 2B 2C
Type
32.27.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
32.27.1b R 1 1 1 1 −1 −1 1 1 −1 −1 −1 1 1 −1
32.27.1c R 1 1 1 1 −1 −1 1 1 −1 −1 1 −1 −1 1
32.27.1d R 1 1 1 1 −1 1 −1 −1 −1 1 −1 −1 1 1
32.27.1e R 1 1 1 1 −1 1 −1 −1 −1 1 1 1 −1 −1
32.27.1f R 1 1 1 1 1 −1 −1 −1 1 −1 −1 1 −1 1
32.27.1g R 1 1 1 1 1 −1 −1 −1 1 −1 1 −1 1 −1
32.27.1h R 1 1 1 1 1 1 1 1 1 1 −1 −1 −1 −1
32.27.2a R 2 −2 −2 2 0 0 −2 2 0 0 0 0 0 0
32.27.2b R 2 −2 −2 2 0 0 2 −2 0 0 0 0 0 0
32.27.2c R 2 −2 2 −2 −2 0 0 0 2 0 0 0 0 0
32.27.2d R 2 −2 2 −2 2 0 0 0 −2 0 0 0 0 0
32.27.2e R 2 2 −2 −2 0 −2 0 0 0 2 0 0 0 0
32.27.2f R 2 2 −2 −2 0 2 0 0 0 −2 0 0 0 0

magma: CharacterTable(G);