Properties

Label 32T39
Order \(32\)
n \(32\)
Cyclic No
Abelian Yes
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^5$

Related objects

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Group action invariants

Degree $n$ :  $32$
Transitive number $t$ :  $39$
Group :  $C_2^5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $1$
Generators:  (1,22)(2,21)(3,24)(4,23)(5,20)(6,19)(7,18)(8,17)(9,14)(10,13)(11,16)(12,15)(25,32)(26,31)(27,30)(28,29), (1,30)(2,29)(3,32)(4,31)(5,10)(6,9)(7,12)(8,11)(13,20)(14,19)(15,18)(16,17)(21,28)(22,27)(23,26)(24,25), (1,6)(2,5)(3,8)(4,7)(9,30)(10,29)(11,32)(12,31)(13,28)(14,27)(15,26)(16,25)(17,24)(18,23)(19,22)(20,21), (1,12)(2,11)(3,10)(4,9)(5,32)(6,31)(7,30)(8,29)(13,24)(14,23)(15,22)(16,21)(17,28)(18,27)(19,26)(20,25), (1,11)(2,12)(3,9)(4,10)(5,31)(6,32)(7,29)(8,30)(13,23)(14,24)(15,21)(16,22)(17,27)(18,28)(19,25)(20,26)
$|\Aut(F/K)|$:  $32$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 31
4:  $C_2^2$ x 155
8:  $C_2^3$ x 155
16:  $C_2^4$ x 31

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 31

Degree 4: $C_2^2$ x 155

Degree 8: $C_2^3$ x 155

Degree 16: $C_2^4$ x 31

Low degree siblings

There are no siblings with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy Classes

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  Yes
Solvable:  Yes
GAP id:  [32, 51]
Character table: Data not available.