Properties

Label 32T36
Order \(32\)
n \(32\)
Cyclic No
Abelian Yes
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2\times C_4^2$

Related objects

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Group action invariants

Degree $n$ :  $32$
Transitive number $t$ :  $36$
Group :  $C_2\times C_4^2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $1$
Generators:  (1,23,11,14)(2,24,12,13)(3,22,10,16)(4,21,9,15)(5,28,30,19)(6,27,29,20)(7,26,31,17)(8,25,32,18), (1,31,12,8)(2,32,11,7)(3,29,9,5)(4,30,10,6)(13,25,23,17)(14,26,24,18)(15,28,22,20)(16,27,21,19), (1,3,2,4)(5,7,6,8)(9,11,10,12)(13,15,14,16)(17,20,18,19)(21,23,22,24)(25,28,26,27)(29,32,30,31)
$|\Aut(F/K)|$:  $32$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_4$ x 12, $C_2^2$ x 7
8:  $C_4\times C_2$ x 18, $C_2^3$
16:  $C_4\times C_2^2$ x 3, $C_4^2$ x 4

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_4$ x 12, $C_2^2$ x 7

Degree 8: $C_4\times C_2$ x 18, $C_2^3$

Degree 16: $C_4\times C_2^2$ x 3, $C_4^2$ x 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 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)(17,20,18,19)(21,23,22,24) (25,28,26,27)(29,32,30,31)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,12,10,11)(13,16,14,15)(17,19,18,20)(21,24,22,23) (25,27,26,28)(29,31,30,32)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1, 5,11,30)( 2, 6,12,29)( 3, 7,10,31)( 4, 8, 9,32)(13,20,24,27)(14,19,23,28) (15,18,21,25)(16,17,22,26)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1, 6,11,29)( 2, 5,12,30)( 3, 8,10,32)( 4, 7, 9,31)(13,19,24,28)(14,20,23,27) (15,17,21,26)(16,18,22,25)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1, 7,12,32)( 2, 8,11,31)( 3, 6, 9,30)( 4, 5,10,29)(13,18,23,26)(14,17,24,25) (15,19,22,27)(16,20,21,28)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1, 8,12,31)( 2, 7,11,32)( 3, 5, 9,29)( 4, 6,10,30)(13,17,23,25)(14,18,24,26) (15,20,22,28)(16,19,21,27)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1, 9, 2,10)( 3,11, 4,12)( 5,32, 6,31)( 7,30, 8,29)(13,22,14,21)(15,24,16,23) (17,28,18,27)(19,25,20,26)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,10, 2, 9)( 3,12, 4,11)( 5,31, 6,32)( 7,29, 8,30)(13,21,14,22)(15,23,16,24) (17,27,18,28)(19,26,20,25)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,11)( 2,12)( 3,10)( 4, 9)( 5,30)( 6,29)( 7,31)( 8,32)(13,24)(14,23)(15,21) (16,22)(17,26)(18,25)(19,28)(20,27)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,12)( 2,11)( 3, 9)( 4,10)( 5,29)( 6,30)( 7,32)( 8,31)(13,23)(14,24)(15,22) (16,21)(17,25)(18,26)(19,27)(20,28)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,13,11,24)( 2,14,12,23)( 3,15,10,21)( 4,16, 9,22)( 5,20,30,27)( 6,19,29,28) ( 7,18,31,25)( 8,17,32,26)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,14,11,23)( 2,13,12,24)( 3,16,10,22)( 4,15, 9,21)( 5,19,30,28)( 6,20,29,27) ( 7,17,31,26)( 8,18,32,25)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,15,12,22)( 2,16,11,21)( 3,14, 9,24)( 4,13,10,23)( 5,18,29,26)( 6,17,30,25) ( 7,19,32,27)( 8,20,31,28)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,16,12,21)( 2,15,11,22)( 3,13, 9,23)( 4,14,10,24)( 5,17,29,25)( 6,18,30,26) ( 7,20,32,28)( 8,19,31,27)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,17, 2,18)( 3,20, 4,19)( 5,22, 6,21)( 7,24, 8,23)( 9,28,10,27)(11,26,12,25) (13,32,14,31)(15,30,16,29)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,18, 2,17)( 3,19, 4,20)( 5,21, 6,22)( 7,23, 8,24)( 9,27,10,28)(11,25,12,26) (13,31,14,32)(15,29,16,30)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,19)( 2,20)( 3,17)( 4,18)( 5,23)( 6,24)( 7,22)( 8,21)( 9,25)(10,26)(11,28) (12,27)(13,29)(14,30)(15,32)(16,31)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,20)( 2,19)( 3,18)( 4,17)( 5,24)( 6,23)( 7,21)( 8,22)( 9,26)(10,25)(11,27) (12,28)(13,30)(14,29)(15,31)(16,32)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,21,12,16)( 2,22,11,15)( 3,23, 9,13)( 4,24,10,14)( 5,25,29,17)( 6,26,30,18) ( 7,28,32,20)( 8,27,31,19)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,22,12,15)( 2,21,11,16)( 3,24, 9,14)( 4,23,10,13)( 5,26,29,18)( 6,25,30,17) ( 7,27,32,19)( 8,28,31,20)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,23,11,14)( 2,24,12,13)( 3,22,10,16)( 4,21, 9,15)( 5,28,30,19)( 6,27,29,20) ( 7,26,31,17)( 8,25,32,18)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,24,11,13)( 2,23,12,14)( 3,21,10,15)( 4,22, 9,16)( 5,27,30,20)( 6,28,29,19) ( 7,25,31,18)( 8,26,32,17)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,25, 2,26)( 3,28, 4,27)( 5,15, 6,16)( 7,14, 8,13)( 9,20,10,19)(11,18,12,17) (21,29,22,30)(23,32,24,31)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,26, 2,25)( 3,27, 4,28)( 5,16, 6,15)( 7,13, 8,14)( 9,19,10,20)(11,17,12,18) (21,30,22,29)(23,31,24,32)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,27)( 2,28)( 3,25)( 4,26)( 5,13)( 6,14)( 7,15)( 8,16)( 9,17)(10,18)(11,20) (12,19)(21,31)(22,32)(23,29)(24,30)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,28)( 2,27)( 3,26)( 4,25)( 5,14)( 6,13)( 7,16)( 8,15)( 9,18)(10,17)(11,19) (12,20)(21,32)(22,31)(23,30)(24,29)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,29,11, 6)( 2,30,12, 5)( 3,32,10, 8)( 4,31, 9, 7)(13,28,24,19)(14,27,23,20) (15,26,21,17)(16,25,22,18)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,30,11, 5)( 2,29,12, 6)( 3,31,10, 7)( 4,32, 9, 8)(13,27,24,20)(14,28,23,19) (15,25,21,18)(16,26,22,17)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,31,12, 8)( 2,32,11, 7)( 3,29, 9, 5)( 4,30,10, 6)(13,25,23,17)(14,26,24,18) (15,28,22,20)(16,27,21,19)$
$ 4, 4, 4, 4, 4, 4, 4, 4 $ $1$ $4$ $( 1,32,12, 7)( 2,31,11, 8)( 3,30, 9, 6)( 4,29,10, 5)(13,26,23,18)(14,25,24,17) (15,27,22,19)(16,28,21,20)$

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  Yes
Solvable:  Yes
GAP id:  [32, 21]
Character table: Data not available.