# Properties

 Label 32T6 Degree $32$ Order $32$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $C_4.C_2^3$

## Group action invariants

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

## 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

Degree 16: $C_2^4$, $(C_2 \times Q_8):C_2$ x 5

## Low degree siblings

16T20 x 5

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

## Group invariants

 Order: $32=2^{5}$ Cyclic: no Abelian: no Solvable: yes GAP id: [32, 50]
 Character table:  2 5 4 5 4 4 4 4 4 4 4 4 4 4 4 4 4 4 1a 2a 2b 4a 4b 4c 4d 4e 4f 4g 2c 4h 2d 4i 2e 4j 2f 2P 1a 1a 1a 2b 2b 2b 2b 2b 2b 2b 1a 2b 1a 2b 1a 2b 1a 3P 1a 2a 2b 4a 4b 4c 4d 4e 4f 4g 2c 4h 2d 4i 2e 4j 2f 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 . . . . . . . . . . . . . .