Properties

Label 32T34
Order \(32\)
n \(32\)
Cyclic No
Abelian Yes
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^3\times C_4$

Related objects

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Group action invariants

Degree $n$ :  $32$
Transitive number $t$ :  $34$
Group :  $C_2^3\times C_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $1$
Generators:  (1,12)(2,11)(3,10)(4,9)(5,30)(6,29)(7,32)(8,31)(13,24)(14,23)(15,22)(16,21)(17,26)(18,25)(19,28)(20,27), (1,18)(2,17)(3,20)(4,19)(5,22)(6,21)(7,24)(8,23)(9,28)(10,27)(11,26)(12,25)(13,32)(14,31)(15,30)(16,29), (1,5,11,29)(2,6,12,30)(3,7,9,31)(4,8,10,32)(13,19,23,27)(14,20,24,28)(15,17,21,25)(16,18,22,26), (1,7,11,31)(2,8,12,32)(3,5,9,29)(4,6,10,30)(13,17,23,25)(14,18,24,26)(15,19,21,27)(16,20,22,28)
$|\Aut(F/K)|$:  $32$

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:  $C_4\times C_2$ x 28, $C_2^3$ x 15
16:  $C_4\times C_2^2$ x 14, $C_2^4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 15

Degree 4: $C_4$ x 8, $C_2^2$ x 35

Degree 8: $C_4\times C_2$ x 28, $C_2^3$ x 15

Degree 16: $C_4\times C_2^2$ x 14, $C_2^4$

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  Yes
Solvable:  Yes
GAP id:  [32, 45]
Character table: Data not available.