Properties

Label 32T22
Order \(32\)
n \(32\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^3.C_4$

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Group action invariants

Degree $n$ :  $32$
Transitive number $t$ :  $22$
Group :  $C_2^3.C_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,17)(2,18)(3,20)(4,19)(5,21)(6,22)(7,24)(8,23)(9,11)(10,12)(13,16)(14,15)(25,28)(26,27)(29,31)(30,32), (1,7,11,31,20,22,25,14)(2,8,12,32,19,21,26,13)(3,5,9,30,17,23,28,16)(4,6,10,29,18,24,27,15)
$|\Aut(F/K)|$:  $32$

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, $(C_8:C_2):C_2$ x 2

Degree 16: $C_2^2 : C_4$, 16T36, 16T41 x 2

Low degree siblings

8T16 x 2, 16T36, 16T41 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 $ $2$ $2$ $( 1, 2)( 3, 4)( 5,24)( 6,23)( 7,21)( 8,22)( 9,10)(11,12)(13,31)(14,32)(15,30) (16,29)(17,18)(19,20)(25,26)(27,28)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,25)(10,26)(11,28)(12,27)(13,30)(14,29)(15,31) (16,32)(17,20)(18,19)(21,23)(22,24)$
$ 8, 8, 8, 8 $ $4$ $8$ $( 1, 5,12,15,20,23,26,29)( 2, 6,11,16,19,24,25,30)( 3, 7,10,13,17,22,27,32) ( 4, 8, 9,14,18,21,28,31)$
$ 8, 8, 8, 8 $ $4$ $8$ $( 1, 7,11,31,20,22,25,14)( 2, 8,12,32,19,21,26,13)( 3, 5, 9,30,17,23,28,16) ( 4, 6,10,29,18,24,27,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,32)( 6,31)( 7,29)( 8,30)(13,23)(14,24)(15,22) (16,21)(17,25)(18,26)(19,27)(20,28)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,11,20,25)( 2,12,19,26)( 3, 9,17,28)( 4,10,18,27)( 5,30,23,16)( 6,29,24,15) ( 7,31,22,14)( 8,32,21,13)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,12,20,26)( 2,11,19,25)( 3,10,17,27)( 4, 9,18,28)( 5,15,23,29)( 6,16,24,30) ( 7,13,22,32)( 8,14,21,31)$
$ 8, 8, 8, 8 $ $4$ $8$ $( 1,13,11, 8,20,32,25,21)( 2,14,12, 7,19,31,26,22)( 3,15, 9, 6,17,29,28,24) ( 4,16,10, 5,18,30,27,23)$
$ 8, 8, 8, 8 $ $4$ $8$ $( 1,15,26, 5,20,29,12,23)( 2,16,25, 6,19,30,11,24)( 3,13,27, 7,17,32,10,22) ( 4,14,28, 8,18,31, 9,21)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,20)( 2,19)( 3,17)( 4,18)( 5,23)( 6,24)( 7,22)( 8,21)( 9,28)(10,27)(11,25) (12,26)(13,32)(14,31)(15,29)(16,30)$

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 7]
Character table:   
      2  5  4  3  3  3  3  4  4  3  3  5

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

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  A -A  1 -1 -1  A -A  1
X.6      1  1 -1 -A  A  1 -1 -1 -A  A  1
X.7      1  1  1  A  A -1 -1 -1 -A -A  1
X.8      1  1  1 -A -A -1 -1 -1  A  A  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