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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 action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $45$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_8:C_2$
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:  (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

|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

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

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:  $3$
Label:  32.43
magma: IdentifyGroup(G);
 
Character table:    
      2  5  3  5  3  4  4  3  3  3  3  4

        1a 2a 2b 4a 4b 2c 8a 2d 8b 2e 4c
     2P 1a 1a 1a 2b 2b 1a 4c 1a 4c 1a 2b
     3P 1a 2a 2b 4a 4b 2c 8a 2d 8b 2e 4c
     5P 1a 2a 2b 4a 4b 2c 8a 2d 8b 2e 4c
     7P 1a 2a 2b 4a 4b 2c 8a 2d 8b 2e 4c

X.1      1  1  1  1  1  1  1  1  1  1  1
X.2      1 -1  1 -1  1  1 -1  1 -1  1  1
X.3      1 -1  1 -1  1  1  1 -1  1 -1  1
X.4      1 -1  1  1 -1 -1 -1  1  1 -1  1
X.5      1 -1  1  1 -1 -1  1 -1 -1  1  1
X.6      1  1  1 -1 -1 -1 -1 -1  1  1  1
X.7      1  1  1 -1 -1 -1  1  1 -1 -1  1
X.8      1  1  1  1  1  1 -1 -1 -1 -1  1
X.9      2  .  2  .  2 -2  .  .  .  . -2
X.10     2  .  2  . -2  2  .  .  .  . -2
X.11     4  . -4  .  .  .  .  .  .  .  .

magma: CharacterTable(G);