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Properties

Label 32T19
Degree $32$
Order $32$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^2.D_4$

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Group action invariants

Degree $n$:  $32$
Transitive number $t$:  $19$
Group:  $C_2^2.D_4$
Parity:  $1$
Primitive:  no
Nilpotency class:  $3$
$|\Aut(F/K)|$:  $32$
Generators:  (1,29,11,7)(2,30,12,8)(3,31,10,6)(4,32,9,5)(13,15,14,16)(17,25,18,26)(19,27,20,28)(21,23,22,24), (1,23,5,27)(2,24,6,28)(3,22,7,25)(4,21,8,26)(9,16,30,20)(10,15,29,19)(11,13,32,18)(12,14,31,17)

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, $C_2^3 : C_4 $ x 2, $C_2^3: C_4$, $C_2^3: C_4$

Degree 16: $C_2^2 : C_4$, 16T33 x 2, 16T52, 16T53

Low degree siblings

8T19 x 2, 8T20, 8T21, 16T33 x 2, 16T52, 16T53

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  no
Abelian:  no
Solvable:  yes
GAP id:  [32, 6]
Character table:    
      2  5  4  3  3  4  4  5  3  3  3  3

        1a 2a 4a 4b 2b 2c 2d 4c 4d 4e 2e
     2P 1a 1a 2a 2a 1a 1a 1a 2b 2b 2d 1a
     3P 1a 2a 4b 4a 2b 2c 2d 4d 4c 4e 2e

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

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