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Properties

Label 32T15
Degree $32$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2\times D_8$

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Show commands: Magma

magma: G := TransitiveGroup(32, 15);
 

Group action invariants

Degree $n$:  $32$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $15$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2\times D_8$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $32$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,19)(2,20)(3,18)(4,17)(5,16)(6,15)(7,13)(8,14)(9,10)(11,12)(21,30)(22,29)(23,31)(24,32)(25,26)(27,28), (1,24)(2,23)(3,21)(4,22)(5,17)(6,18)(7,20)(8,19)(9,14)(10,13)(11,15)(12,16)(25,30)(26,29)(27,32)(28,31), (1,11,20,26)(2,12,19,25)(3,9,17,28)(4,10,18,27)(5,14,21,31)(6,13,22,32)(7,15,24,29)(8,16,23,30)
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, $C_2^3$
$16$:  $D_{8}$ x 2, $D_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7, $D_{4}$ x 4

Degree 8: $C_2^3$, $D_4$ x 2, $D_{8}$ x 4, $D_4\times C_2$ x 4

Degree 16: $D_4\times C_2$, $D_{8}$ x 2, 16T29 x 4

Low degree siblings

16T29 x 4

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 2)( 3, 4)( 5,30)( 6,29)( 7,32)( 8,31)( 9,27)(10,28)(11,25)(12,26)(13,24) (14,23)(15,22)(16,21)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,24) (22,23)(25,27)(26,28)(29,31)(30,32)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 4)( 2, 3)( 5,32)( 6,31)( 7,30)( 8,29)( 9,25)(10,26)(11,27)(12,28)(13,21) (14,22)(15,23)(16,24)(17,19)(18,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,28) (16,27)(17,24)(18,23)(19,22)(20,21)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1, 6, 9,16,20,22,28,30)( 2, 5,10,15,19,21,27,29)( 3, 8,11,13,17,23,26,32) ( 4, 7,12,14,18,24,25,31)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,31)(10,32)(11,29)(12,30)(13,27)(14,28)(15,26) (16,25)(17,21)(18,22)(19,23)(20,24)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1, 8, 9,13,20,23,28,32)( 2, 7,10,14,19,24,27,31)( 3, 6,11,16,17,22,26,30) ( 4, 5,12,15,18,21,25,29)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 9,20,28)( 2,10,19,27)( 3,11,17,26)( 4,12,18,25)( 5,15,21,29)( 6,16,22,30) ( 7,14,24,31)( 8,13,23,32)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,11,20,26)( 2,12,19,25)( 3, 9,17,28)( 4,10,18,27)( 5,14,21,31)( 6,13,22,32) ( 7,15,24,29)( 8,16,23,30)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1,13,28, 8,20,32, 9,23)( 2,14,27, 7,19,31,10,24)( 3,16,26, 6,17,30,11,22) ( 4,15,25, 5,18,29,12,21)$
$ 8, 8, 8, 8 $ $2$ $8$ $( 1,16,28, 6,20,30, 9,22)( 2,15,27, 5,19,29,10,21)( 3,13,26, 8,17,32,11,23) ( 4,14,25, 7,18,31,12,24)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,17)( 2,18)( 3,20)( 4,19)( 5,24)( 6,23)( 7,21)( 8,22)( 9,26)(10,25)(11,28) (12,27)(13,30)(14,29)(15,31)(16,32)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,20)( 2,19)( 3,17)( 4,18)( 5,21)( 6,22)( 7,24)( 8,23)( 9,28)(10,27)(11,26) (12,25)(13,32)(14,31)(15,29)(16,30)$

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:  32.39
magma: IdentifyGroup(G);
 
Character table:    
      2  5  3  5  3  3  4  3  4  4  4  4  4  5  5

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

X.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
X.3      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
X.5      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
X.7      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
X.9      2  . -2  .  .  .  .  . -2  2  .  . -2  2
X.10     2  .  2  .  .  .  .  . -2 -2  .  .  2  2
X.11     2  . -2  .  .  A  . -A  .  .  A -A  2 -2
X.12     2  . -2  .  . -A  .  A  .  . -A  A  2 -2
X.13     2  .  2  .  .  A  .  A  .  . -A -A -2 -2
X.14     2  .  2  .  . -A  . -A  .  .  A  A -2 -2

A = -E(8)+E(8)^3
  = -Sqrt(2) = -r2

magma: CharacterTable(G);