# Properties

 Label 32T21 Degree $32$ Order $32$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $C_8:C_2^2$

## Group action invariants

 Degree $n$: $32$ Transitive number $t$: $21$ Group: $C_8:C_2^2$ Parity: $1$ Primitive: no Nilpotency class: $3$ $|\Aut(F/K)|$: $32$ Generators: (1,23,28,13,18,8,12,29)(2,24,27,14,17,7,11,30)(3,21,25,15,19,6,9,32)(4,22,26,16,20,5,10,31), (1,19)(2,20)(3,18)(4,17)(5,14)(6,13)(7,16)(8,15)(9,12)(10,11)(21,29)(22,30)(23,32)(24,31)(25,28)(26,27), (1,7,12,14,18,24,28,30)(2,8,11,13,17,23,27,29)(3,5,9,16,19,22,25,31)(4,6,10,15,20,21,26,32)

## 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, $C_2^3$
$16$:  $D_4\times C_2$

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, $Z_8 : Z_8^\times$ x 2

Degree 16: $D_4\times C_2$, 16T35, 16T38 x 2, 16T45

## Low degree siblings

8T15 x 2, 16T35, 16T38 x 2, 16T45

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

## Group invariants

 Order: $32=2^{5}$ Cyclic: no Abelian: no Solvable: yes GAP id: [32, 43]
 Character table:  2 5 4 3 3 3 3 3 3 4 4 5 1a 2a 2b 2c 2d 4a 8a 8b 4b 4c 2e 2P 1a 1a 1a 1a 1a 2e 4c 4c 2e 2e 1a 3P 1a 2a 2b 2c 2d 4a 8a 8b 4b 4c 2e 5P 1a 2a 2b 2c 2d 4a 8a 8b 4b 4c 2e 7P 1a 2a 2b 2c 2d 4a 8a 8b 4b 4c 2e X.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 X.3 1 -1 -1 1 1 -1 -1 1 -1 1 1 X.4 1 -1 1 -1 -1 1 -1 1 -1 1 1 X.5 1 -1 1 -1 1 -1 1 -1 -1 1 1 X.6 1 1 -1 -1 -1 -1 1 1 1 1 1 X.7 1 1 -1 -1 1 1 -1 -1 1 1 1 X.8 1 1 1 1 -1 -1 -1 -1 1 1 1 X.9 2 2 . . . . . . -2 -2 2 X.10 2 -2 . . . . . . 2 -2 2 X.11 4 . . . . . . . . . -4