Properties

Label 32T43
Order \(32\)
n \(32\)
Cyclic No
Abelian Yes
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_4\times C_8$

Related objects

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Group action invariants

Degree $n$ :  $32$
Transitive number $t$ :  $43$
Group :  $C_4\times C_8$
Parity:  $1$
Primitive:  No
Nilpotency class:  $1$
Generators:  (1,5,12,31,2,6,11,32)(3,8,9,30,4,7,10,29)(13,17,24,27,14,18,23,28)(15,20,21,26,16,19,22,25), (1,18,3,19)(2,17,4,20)(5,23,8,22)(6,24,7,21)(9,25,12,28)(10,26,11,27)(13,30,15,31)(14,29,16,32)
$|\Aut(F/K)|$:  $32$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 6, $C_2^2$
8:  $C_8$ x 4, $C_4\times C_2$ x 3
16:  $C_4^2$, $C_8\times C_2$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 6, $C_2^2$

Degree 8: $C_8$ x 4, $C_4\times C_2$ x 3

Degree 16: $C_4^2$, $C_8\times C_2$ x 2

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  Yes
Solvable:  Yes
GAP id:  [32, 3]
Character table: Data not available.