# Properties

 Label 32T7 Degree $32$ Order $32$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $C_2\times C_2^2:C_4$

## Group action invariants

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

## 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
$8$:  $D_{4}$ x 4, $C_4\times C_2$ x 6, $C_2^3$
$16$:  $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $C_4\times C_2^2$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_4$ x 4, $C_2^2$ x 7, $D_{4}$ x 8

Degree 8: $C_4\times C_2$ x 6, $C_2^3$, $D_4$ x 4, $D_4\times C_2$ x 8, $C_2^2:C_4$ x 8

Degree 16: $C_4\times C_2^2$, $D_4\times C_2$ x 2, $C_2^2 : C_4$ x 4, $C_2 \times (C_2^2:C_4)$ x 4

## Low degree siblings

16T21 x 4

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

## Group invariants

 Order: $32=2^{5}$ Cyclic: no Abelian: no Solvable: yes GAP id: [32, 22]
 Character table:  2 5 4 4 5 4 5 5 4 5 5 4 4 4 4 4 4 4 4 5 5 1a 2a 2b 2c 2d 2e 2f 2g 2h 2i 4a 4b 4c 4d 4e 4f 4g 4h 2j 2k 2P 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 2c 2k 2k 2c 2c 2k 2c 2k 1a 1a 3P 1a 2a 2b 2c 2d 2e 2f 2g 2h 2i 4d 4c 4b 4a 4g 4h 4e 4f 2j 2k 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 -1 1 -1 1 -1 1 A -A A -A -A A A -A 1 -1 X.10 1 -1 1 -1 -1 1 -1 1 -1 1 -A A -A A A -A -A A 1 -1 X.11 1 -1 1 -1 1 -1 1 -1 1 -1 A -A A -A A -A -A A 1 -1 X.12 1 -1 1 -1 1 -1 1 -1 1 -1 -A A -A A -A A A -A 1 -1 X.13 1 1 -1 -1 -1 -1 1 1 1 -1 A A -A -A A A -A -A 1 -1 X.14 1 1 -1 -1 -1 -1 1 1 1 -1 -A -A A A -A -A A A 1 -1 X.15 1 1 -1 -1 1 1 -1 -1 -1 1 A A -A -A -A -A A A 1 -1 X.16 1 1 -1 -1 1 1 -1 -1 -1 1 -A -A A A A A -A -A 1 -1 X.17 2 . . -2 . -2 2 . -2 2 . . . . . . . . -2 2 X.18 2 . . -2 . 2 -2 . 2 -2 . . . . . . . . -2 2 X.19 2 . . 2 . -2 -2 . 2 2 . . . . . . . . -2 -2 X.20 2 . . 2 . 2 2 . -2 -2 . . . . . . . . -2 -2 A = -E(4) = -Sqrt(-1) = -i