Properties

Label 32T5
Order \(32\)
n \(32\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_4\times D_4$

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Group action invariants

Degree $n$ :  $32$
Transitive number $t$ :  $5$
Group :  $C_4\times D_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,6,3,8)(2,5,4,7)(9,29,12,31)(10,30,11,32)(13,22,16,23)(14,21,15,24)(17,25,20,28)(18,26,19,27), (1,23,3,22)(2,24,4,21)(5,20,7,17)(6,19,8,18)(9,16,12,13)(10,15,11,14)(25,32,28,30)(26,31,27,29), (1,4)(2,3)(5,8)(6,7)(9,11)(10,12)(13,17)(14,18)(15,19)(16,20)(21,26)(22,25)(23,28)(24,27)(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:  $C_4$ x 4, $C_2^2$ x 7
8:  $D_{4}$ x 2, $C_4\times C_2$ x 6, $C_2^3$
16:  $D_4\times C_2$, $Q_8:C_2$, $C_4\times C_2^2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_4$ x 4, $C_2^2$ x 7, $D_{4}$ x 4

Degree 8: $C_4\times C_2$ x 6, $C_2^3$, $D_4$ x 2, $D_4\times C_2$ x 4, $Q_8:C_2$ x 3

Degree 16: $C_4\times C_2^2$, $D_4\times C_2$, $Q_8 : C_2$, $C_4 \times D_4$ x 4

Low degree siblings

16T19 x 4

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 25]
Character table:   
      2  5  4  5  4  5  4  5  4  5  5  4  4  4  4  4  4  4  4  5  5

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

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  A -A -A  A -A  A -1  1 -1  1 -A  A -A  A  1 -1
X.10     1 -1 -1  1 -A  A  A -A  A -A -1  1 -1  1  A -A  A -A  1 -1
X.11     1 -1 -1  1  A -A -A  A -A  A  1 -1  1 -1  A -A  A -A  1 -1
X.12     1 -1 -1  1 -A  A  A -A  A -A  1 -1  1 -1 -A  A -A  A  1 -1
X.13     1  1 -1 -1  A  A -A -A -A  A -1 -1  1  1 -A -A  A  A  1 -1
X.14     1  1 -1 -1 -A -A  A  A  A -A -1 -1  1  1  A  A -A -A  1 -1
X.15     1  1 -1 -1  A  A -A -A -A  A  1  1 -1 -1  A  A -A -A  1 -1
X.16     1  1 -1 -1 -A -A  A  A  A -A  1  1 -1 -1 -A -A  A  A  1 -1
X.17     2  .  2  . -2  . -2  .  2  2  .  .  .  .  .  .  .  . -2 -2
X.18     2  .  2  .  2  .  2  . -2 -2  .  .  .  .  .  .  .  . -2 -2
X.19     2  . -2  .  B  . -B  .  B -B  .  .  .  .  .  .  .  . -2  2
X.20     2  . -2  . -B  .  B  . -B  B  .  .  .  .  .  .  .  . -2  2

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