# Properties

 Label 32T3 Degree $32$ Order $32$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $(C_2\times C_4):C_4$

## Group action invariants

 Degree $n$: $32$ Transitive number $t$: $3$ Group: $(C_2\times C_4):C_4$ Parity: $1$ Primitive: no Nilpotency class: $2$ $|\Aut(F/K)|$: $32$ 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)

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

## 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