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Properties

Label 32T23
Order \(32\)
n \(32\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $(C_2^2\times C_4):C_2$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $D_{4}$ x 2, $C_2^3$
16:  $D_4\times C_2$, $Q_8:C_2$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

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

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

Degree 16: $D_4\times C_2$, $Q_8 : C_2$ x 2, 16T37, 16T54 x 2

Low degree siblings

16T37, 16T54 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 $ $4$ $2$ $( 1, 2)( 3, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,18)(14,17)(15,19)(16,20)(21,28) (22,27)(23,25)(24,26)(29,30)(31,32)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,19)(18,20)(21,24) (22,23)(25,27)(26,28)(29,32)(30,31)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,31)(10,32)(11,29)(12,30)(13,25)(14,26)(15,28) (16,27)(17,21)(18,22)(19,24)(20,23)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $4$ $4$ $( 1, 6, 4, 7)( 2, 5, 3, 8)( 9,29,12,32)(10,30,11,31)(13,22,16,23)(14,21,15,24) (17,26,19,28)(18,25,20,27)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,31)( 6,32)( 7,29)( 8,30)(13,21)(14,22)(15,23) (16,24)(17,25)(18,26)(19,27)(20,28)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,13, 4,16)( 2,14, 3,15)( 5,25, 8,27)( 6,26, 7,28)( 9,21,12,24)(10,22,11,23) (17,30,19,31)(18,29,20,32)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $4$ $4$ $( 1,14,30,20)( 2,13,29,19)( 3,16,32,17)( 4,15,31,18)( 5,28,12,22)( 6,27,11,21) ( 7,25,10,24)( 8,26, 9,23)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,16, 4,13)( 2,15, 3,14)( 5,27, 8,25)( 6,28, 7,26)( 9,24,12,21)(10,23,11,22) (17,31,19,30)(18,32,20,29)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,21, 4,24)( 2,22, 3,23)( 5,17, 8,19)( 6,18, 7,20)( 9,13,12,16)(10,14,11,15) (25,30,27,31)(26,29,28,32)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $4$ $4$ $( 1,22,31,26)( 2,21,32,25)( 3,24,29,27)( 4,23,30,28)( 5,20, 9,15)( 6,19,10,16) ( 7,17,11,13)( 8,18,12,14)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1,24, 4,21)( 2,23, 3,22)( 5,19, 8,17)( 6,20, 7,18)( 9,16,12,13)(10,15,11,14) (25,31,27,30)(26,32,28,29)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,30)( 2,29)( 3,32)( 4,31)( 5,12)( 6,11)( 7,10)( 8, 9)(13,19)(14,20)(15,18) (16,17)(21,27)(22,28)(23,26)(24,25)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,31)( 2,32)( 3,29)( 4,30)( 5, 9)( 6,10)( 7,11)( 8,12)(13,17)(14,18)(15,20) (16,19)(21,25)(22,26)(23,28)(24,27)$

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 30]
Character table:    
      2  5  3  5  4  3  4  4  3  4  4  3  4  5  5

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

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

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