# Properties

 Label 32T9 Degree $32$ Order $32$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $Q_8:C_2^2$

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## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 15
$4$:  $C_2^2$ x 35
$8$:  $C_2^3$ x 15
$16$:  $C_2^4$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 15

Degree 4: $C_2^2$ x 35

Degree 8: $C_2^3$ x 15, $C_2^3 : D_4$ x 6

Degree 16: $C_2^4$, $Q_8 : C_2^2$ x 9

## Low degree siblings

8T22 x 6, 16T23 x 9

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

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

## Group invariants

 Order: $32=2^{5}$ Cyclic: no Abelian: no Solvable: yes GAP id: [32, 49]
 Character table:  2 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 5 1a 2a 2b 2c 2d 4a 4b 2e 2f 2g 4c 4d 2h 4e 4f 2i 2j 2P 1a 1a 1a 1a 1a 2j 2j 1a 1a 1a 2j 2j 1a 2j 2j 1a 1a 3P 1a 2a 2b 2c 2d 4a 4b 2e 2f 2g 4c 4d 2h 4e 4f 2i 2j X.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 X.3 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 X.5 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 X.7 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 X.9 1 -1 1 -1 1 -1 -1 1 1 -1 1 -1 -1 1 -1 1 1 X.10 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 1 -1 -1 1 X.11 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 1 1 X.12 1 1 -1 -1 1 1 -1 -1 -1 -1 1 1 -1 -1 1 1 1 X.13 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 1 -1 -1 1 X.14 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 1 1 1 1 X.15 1 1 1 1 -1 -1 -1 -1 1 1 1 1 -1 -1 -1 -1 1 X.16 1 1 1 1 1 1 1 1 -1 -1 -1 -1 -1 -1 -1 -1 1 X.17 4 . . . . . . . . . . . . . . . -4