Properties

Label 32T13
Order \(32\)
n \(32\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_4^2:C_2$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $C_2^3$
16:  $Q_8:C_2$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7

Degree 8: $C_2^3$, $Q_8:C_2$ x 9

Degree 16: $Q_8 : C_2$ x 3, 16T27

Low degree siblings

16T27

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 33]
Character table:   
      2  5  3  5  4  3  4  4  3  3  4  4  5  5  4

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

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 -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
X.7      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
X.9      2  . -2  A  . -A  .  .  .  .  . -2  2  .
X.10     2  . -2 -A  .  A  .  .  .  .  . -2  2  .
X.11     2  . -2  .  .  .  .  .  .  A -A  2 -2  .
X.12     2  . -2  .  .  .  .  .  . -A  A  2 -2  .
X.13     2  .  2  .  .  .  A  .  .  .  . -2 -2 -A
X.14     2  .  2  .  .  . -A  .  .  .  . -2 -2  A

A = -2*E(4)
  = -2*Sqrt(-1) = -2i