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Properties

Label 32T25
Degree $32$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_4.D_4$

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Show commands: Magma

magma: G := TransitiveGroup(32, 25);
 

Group action invariants

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

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, $C_2^2:C_4$ x 2

Degree 16: $C_2^2 : C_4$, 16T40

Low degree siblings

16T40

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

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:  32.8
magma: IdentifyGroup(G);
 
Character table:    
      2  5  4  5  4  4  3  3  3  3  3  3

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

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

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

magma: CharacterTable(G);