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Properties

Label 32T4
Degree $32$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $D_4:C_2^2$

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Show commands: Magma

magma: G := TransitiveGroup(32, 4);
 

Group action invariants

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

Degree 4: $C_2^2$ x 35

Degree 8: $C_2^3$ x 15, $Q_8:C_2$ x 6

Degree 16: $C_2^4$, $Q_8 : C_2$ x 2, $C_2 \times (C_4\times C_2):C_2$ x 6

Low degree siblings

16T18 x 6

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

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  4  5  4  4  4  4  5  5  5  5

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

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);