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Properties

Label 32T12
Degree $32$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $D_4:C_4$

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Show commands: Magma

magma: G := TransitiveGroup(32, 12);
 

Group action invariants

Degree $n$:  $32$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $12$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_4:C_4$
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:  (1,7,3,5)(2,8,4,6)(9,30,11,31)(10,29,12,32)(13,28,16,25)(14,27,15,26)(17,23,19,21)(18,24,20,22), (1,13,27,6,18,32,9,21)(2,14,28,5,17,31,10,22)(3,16,26,8,20,29,11,23)(4,15,25,7,19,30,12,24)
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, $C_4\times C_2$
$16$:  $D_{8}$, $QD_{16}$, $C_2^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$, $D_{4}$ x 4

Degree 8: $C_4\times C_2$, $D_4$ x 2, $D_{8}$ x 2, $QD_{16}$, $C_2^2:C_4$ x 2

Degree 16: $C_2^2 : C_4$, $QD_{16}$, $D_{8}$, 16T26 x 2

Low degree siblings

16T26 x 2

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

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:  32.9
magma: IdentifyGroup(G);
 
Character table:    
      2  5  3  5  3  3  4  3  4  4  4  4  4  5  5

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

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  A -A -A  A -1  1  A -A  1 -1
X.6      1 -1 -1  1 -A  A  A -A -1  1 -A  A  1 -1
X.7      1  1 -1 -1  A  A -A -A -1  1 -A  A  1 -1
X.8      1  1 -1 -1 -A -A  A  A -1  1  A -A  1 -1
X.9      2  . -2  .  .  .  .  .  2 -2  .  .  2 -2
X.10     2  .  2  .  .  .  .  . -2 -2  .  .  2  2
X.11     2  .  2  .  .  B  .  B  .  . -B -B -2 -2
X.12     2  .  2  .  . -B  . -B  .  .  B  B -2 -2
X.13     2  . -2  .  .  C  . -C  .  .  C -C -2  2
X.14     2  . -2  .  . -C  .  C  .  . -C  C -2  2

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(8)+E(8)^3
  = -Sqrt(2) = -r2
C = E(8)+E(8)^3
  = Sqrt(-2) = i2

magma: CharacterTable(G);