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Properties

Label 32T17
Order \(32\)
n \(32\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^2:Q_8$

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Group action invariants

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

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, $C_2^3$, $Q_8$ x 2
16:  $D_4\times C_2$, $Q_8:C_2$, $D_8$

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, $Q_8$ x 2, $D_4\times C_2$ x 4, $Q_8:C_2$ x 3

Degree 16: $D_8$, $D_4\times C_2$, $Q_8 : C_2$, 16T31 x 2

Low degree siblings

16T31 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 $ $2$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,19)(14,20)(15,17)(16,18)(21,27) (22,28)(23,26)(24,25)(29,30)(31,32)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,17)(14,18)(15,19)(16,20)(21,25) (22,26)(23,28)(24,27)(29,31)(30,32)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,24) (22,23)(25,27)(26,28)(29,32)(30,31)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 5, 4, 7)( 2, 6, 3, 8)( 9,29,11,32)(10,30,12,31)(13,27,15,25)(14,28,16,26) (17,24,19,21)(18,23,20,22)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 6, 4, 8)( 2, 5, 3, 7)( 9,30,11,31)(10,29,12,32)(13,21,15,24)(14,22,16,23) (17,25,19,27)(18,26,20,28)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 8, 4, 6)( 2, 7, 3, 5)( 9,31,11,30)(10,32,12,29)(13,24,15,21)(14,23,16,22) (17,27,19,25)(18,28,20,26)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $2$ $4$ $( 1, 9, 4,11)( 2,10, 3,12)( 5,29, 7,32)( 6,30, 8,31)(13,23,15,22)(14,24,16,21) (17,28,19,26)(18,27,20,25)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $4$ $4$ $( 1,13,32,18)( 2,14,31,17)( 3,16,30,19)( 4,15,29,20)( 5,25, 9,22)( 6,26,10,21) ( 7,27,11,23)( 8,28,12,24)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $4$ $4$ $( 1,14, 4,16)( 2,13, 3,15)( 5,26, 7,28)( 6,25, 8,27)( 9,21,11,24)(10,22,12,23) (17,29,19,32)(18,30,20,31)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $4$ $4$ $( 1,21, 4,24)( 2,22, 3,23)( 5,19, 7,17)( 6,20, 8,18)( 9,16,11,14)(10,15,12,13) (25,30,27,31)(26,29,28,32)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $4$ $4$ $( 1,22,32,25)( 2,21,31,26)( 3,24,30,28)( 4,23,29,27)( 5,20, 9,15)( 6,19,10,16) ( 7,18,11,13)( 8,17,12,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,29)( 2,30)( 3,31)( 4,32)( 5,11)( 6,12)( 7, 9)( 8,10)(13,20)(14,19)(15,18) (16,17)(21,28)(22,27)(23,25)(24,26)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,32)( 2,31)( 3,30)( 4,29)( 5, 9)( 6,10)( 7,11)( 8,12)(13,18)(14,17)(15,20) (16,19)(21,26)(22,25)(23,27)(24,28)$

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 29]
Character table:    
      2  5  4  4  5  4  4  4  4  3  3  3  3  5  5

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

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 -2  .  .  2  .  .  .  . -2 -2
X.12     2  .  .  2  2  .  . -2  .  .  .  . -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