# Properties

 Label 32T262 Degree $32$ Order $64$ Cyclic no Abelian no Solvable yes Primitive no $p$-group yes Group: $C_2^2\times C_2^2:C_4$

# Related objects

## Group action invariants

 Degree $n$: $32$ Transitive number $t$: $262$ Group: $C_2^2\times C_2^2:C_4$ Parity: $1$ Primitive: no Nilpotency class: $2$ $|\Aut(F/K)|$: $16$ Generators: (1,11)(2,12)(3,9)(4,10)(5,29)(6,30)(7,31)(8,32)(13,25)(14,26)(15,27)(16,28)(17,21)(18,22)(19,23)(20,24), (1,32)(2,31)(3,30)(4,29)(5,10)(6,9)(7,12)(8,11)(13,18)(14,17)(15,20)(16,19)(21,26)(22,25)(23,28)(24,27), (1,27,31,23)(2,28,32,24)(3,25,29,21)(4,26,30,22)(5,15,9,19)(6,16,10,20)(7,13,11,17)(8,14,12,18), (1,9)(2,10)(3,11)(4,12)(5,31)(6,32)(7,29)(8,30)(13,21)(14,22)(15,23)(16,24)(17,25)(18,26)(19,27)(20,28)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 15
$4$:  $C_4$ x 8, $C_2^2$ x 35
$8$:  $D_{4}$ x 8, $C_4\times C_2$ x 28, $C_2^3$ x 15
$16$:  $D_4\times C_2$ x 12, $C_2^2:C_4$ x 16, $C_4\times C_2^2$ x 14, $C_2^4$
$32$:  $C_2 \times (C_2^2:C_4)$ x 12, $C_2^2 \times D_4$ x 2, 32T34

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, $C_2^3$, $D_4\times C_2$ x 12, $C_2^2:C_4$ x 16

Degree 16: $C_4\times C_2^2$, $C_2 \times (C_2^2:C_4)$ x 12, $C_2^2 \times D_4$ x 2

## Low degree siblings

32T262 x 7

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

## Group invariants

 Order: $64=2^{6}$ Cyclic: no Abelian: no Solvable: yes GAP id: [64, 193]
 Character table: not available.