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Properties

Label 32T3
Order \(32\)
n \(32\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $(C_2\times C_4):C_4$

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Group action invariants

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

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:  $C_4\times C_2$ x 6, $C_2^3$
16:  $Q_8:C_2$ x 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

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

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

Low degree siblings

16T17 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 $ $2$ $2$ $( 1, 2)( 3, 4)( 5,22)( 6,21)( 7,24)( 8,23)( 9,10)(11,12)(13,30)(14,29)(15,32) (16,31)(17,18)(19,20)(25,26)(27,28)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)(17,20)(18,19)(21,24) (22,23)(25,28)(26,27)(29,32)(30,31)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 4)( 2, 3)( 5,23)( 6,24)( 7,21)( 8,22)( 9,12)(10,11)(13,31)(14,32)(15,29) (16,30)(17,19)(18,20)(25,27)(26,28)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 5, 3, 8)( 2, 6, 4, 7)( 9,15,11,14)(10,16,12,13)(17,24,20,21)(18,23,19,22) (25,32,28,29)(26,31,27,30)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 6,17,23)( 2, 5,18,24)( 3, 7,20,22)( 4, 8,19,21)( 9,16,27,29)(10,15,28,30) (11,13,26,32)(12,14,25,31)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 7,17,22)( 2, 8,18,21)( 3, 6,20,23)( 4, 5,19,24)( 9,13,27,32)(10,14,28,31) (11,16,26,29)(12,15,25,30)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 8, 3, 5)( 2, 7, 4, 6)( 9,14,11,15)(10,13,12,16)(17,21,20,24)(18,22,19,23) (25,29,28,32)(26,30,27,31)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1, 9,20,26)( 2,10,19,25)( 3,11,17,27)( 4,12,18,28)( 5,15,21,31)( 6,16,22,32) ( 7,13,23,29)( 8,14,24,30)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,10,20,25)( 2, 9,19,26)( 3,12,17,28)( 4,11,18,27)( 5,32,21,16)( 6,31,22,15) ( 7,30,23,14)( 8,29,24,13)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,11,20,27)( 2,12,19,28)( 3, 9,17,26)( 4,10,18,25)( 5,14,21,30)( 6,13,22,29) ( 7,16,23,32)( 8,15,24,31)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,12,20,28)( 2,11,19,27)( 3,10,17,25)( 4, 9,18,26)( 5,29,21,13)( 6,30,22,14) ( 7,31,23,15)( 8,32,24,16)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,13, 3,16)( 2,14, 4,15)( 5,25, 8,28)( 6,26, 7,27)( 9,23,11,22)(10,24,12,21) (17,32,20,29)(18,31,19,30)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,14,17,31)( 2,13,18,32)( 3,15,20,30)( 4,16,19,29)( 5, 9,24,27)( 6,10,23,28) ( 7,12,22,25)( 8,11,21,26)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,15,17,30)( 2,16,18,29)( 3,14,20,31)( 4,13,19,32)( 5,11,24,26)( 6,12,23,25) ( 7,10,22,28)( 8, 9,21,27)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,16, 3,13)( 2,15, 4,14)( 5,28, 8,25)( 6,27, 7,26)( 9,22,11,23)(10,21,12,24) (17,29,20,32)(18,30,19,31)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,17)( 2,18)( 3,20)( 4,19)( 5,24)( 6,23)( 7,22)( 8,21)( 9,27)(10,28)(11,26) (12,25)(13,32)(14,31)(15,30)(16,29)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,20)( 2,19)( 3,17)( 4,18)( 5,21)( 6,22)( 7,23)( 8,24)( 9,26)(10,25)(11,27) (12,28)(13,29)(14,30)(15,31)(16,32)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,26,20, 9)( 2,25,19,10)( 3,27,17,11)( 4,28,18,12)( 5,31,21,15)( 6,32,22,16) ( 7,29,23,13)( 8,30,24,14)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,27,20,11)( 2,28,19,12)( 3,26,17, 9)( 4,25,18,10)( 5,30,21,14)( 6,29,22,13) ( 7,32,23,16)( 8,31,24,15)$

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 24]
Character table:    
      2  5  4  5  4  4  4  4  4  5  4  5  4  4  4  4  4  5  5  5  5

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

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

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