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Properties

Label 16T18
Degree $16$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2 \times (C_4\times C_2):C_2$

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Show commands: Magma

magma: G := TransitiveGroup(16, 18);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $18$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2 \times (C_4\times C_2):C_2$
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:  (1,16)(2,15)(3,6)(4,5)(7,10)(8,9)(11,14)(12,13), (1,5,10,13)(2,6,9,14)(3,8,11,15)(4,7,12,16), (1,9)(2,10)(3,4)(5,14)(6,13)(7,8)(11,12)(15,16), (1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 15
$4$:  $C_2^2$ x 35
$8$:  $C_2^3$ x 15
$16$:  $Q_8:C_2$ x 2, $C_2^4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7

Degree 8: $C_2^3$, $Q_8:C_2$ x 2

Low degree siblings

16T18 x 5, 32T4

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

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:  32.48
magma: IdentifyGroup(G);
 
Character table:    
      2  5  4  5  4  4  4  4  4  5  4  5  4  4  4  4  4  5  5  5  5

        1a 2a 2b 2c 2d 4a 2e 4b 4c 4d 4e 4f 4g 2f 4h 2g 2h 2i 4i 4j
     2P 1a 1a 1a 1a 1a 2i 1a 2i 2i 2i 2i 2i 2i 1a 2i 1a 1a 1a 2i 2i
     3P 1a 2a 2b 2c 2d 4a 2e 4b 4i 4d 4j 4f 4g 2f 4h 2g 2h 2i 4c 4e

X.1      1  1  1  1  1  1  1  1  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 -1 -1  1  1 -1 -1  1 -1  1
X.3      1 -1 -1  1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1
X.4      1 -1 -1  1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1 -1  1
X.5      1 -1 -1  1  1 -1 -1  1  1 -1 -1  1 -1  1  1 -1 -1  1  1 -1
X.6      1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1 -1  1 -1  1  1 -1 -1
X.7      1 -1  1 -1 -1  1 -1  1  1 -1  1 -1 -1  1 -1  1  1  1  1  1
X.8      1 -1  1 -1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1  1  1 -1 -1
X.9      1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1 -1  1  1  1  1
X.10     1  1 -1 -1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1  1 -1  1
X.11     1  1 -1 -1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1 -1  1  1 -1
X.12     1  1 -1 -1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1  1 -1  1
X.13     1  1 -1 -1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1  1  1 -1
X.14     1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1 -1 -1
X.15     1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1
X.16     1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1 -1 -1
X.17     2  . -2  .  .  .  .  .  A  . -A  .  .  .  .  .  2 -2 -A  A
X.18     2  . -2  .  .  .  .  . -A  .  A  .  .  .  .  .  2 -2  A -A
X.19     2  .  2  .  .  .  .  .  A  .  A  .  .  .  .  . -2 -2 -A -A
X.20     2  .  2  .  .  .  .  . -A  . -A  .  .  .  .  . -2 -2  A  A

A = -2*E(4)
  = -2*Sqrt(-1) = -2i

magma: CharacterTable(G);