Properties

Label 32T21
Order \(32\)
n \(32\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_8:C_2^2$

Learn more about

Group action invariants

Degree $n$ :  $32$
Transitive number $t$ :  $21$
Group :  $C_8:C_2^2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,23,28,13,18,8,12,29)(2,24,27,14,17,7,11,30)(3,21,25,15,19,6,9,32)(4,22,26,16,20,5,10,31), (1,19)(2,20)(3,18)(4,17)(5,14)(6,13)(7,16)(8,15)(9,12)(10,11)(21,29)(22,30)(23,32)(24,31)(25,28)(26,27), (1,7,12,14,18,24,28,30)(2,8,11,13,17,23,27,29)(3,5,9,16,19,22,25,31)(4,6,10,15,20,21,26,32)
$|\Aut(F/K)|$:  $32$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $D_{4}$ x 2, $C_2^3$
16:  $D_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7, $D_{4}$ x 4

Degree 8: $C_2^3$, $D_4$ x 2, $D_4\times C_2$ x 4, $Z_8 : Z_8^\times$ x 2

Degree 16: $D_4\times C_2$, 16T35, 16T38 x 2, 16T45

Low degree siblings

8T15 x 2, 16T35, 16T38 x 2, 16T45

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 43]
Character table:   
      2  5  4  3  3  3  3  3  3  4  4  5

        1a 2a 2b 2c 2d 4a 8a 8b 4b 4c 2e
     2P 1a 1a 1a 1a 1a 2e 4c 4c 2e 2e 1a
     3P 1a 2a 2b 2c 2d 4a 8a 8b 4b 4c 2e
     5P 1a 2a 2b 2c 2d 4a 8a 8b 4b 4c 2e
     7P 1a 2a 2b 2c 2d 4a 8a 8b 4b 4c 2e

X.1      1  1  1  1  1  1  1  1  1  1  1
X.2      1 -1 -1  1 -1  1  1 -1 -1  1  1
X.3      1 -1 -1  1  1 -1 -1  1 -1  1  1
X.4      1 -1  1 -1 -1  1 -1  1 -1  1  1
X.5      1 -1  1 -1  1 -1  1 -1 -1  1  1
X.6      1  1 -1 -1 -1 -1  1  1  1  1  1
X.7      1  1 -1 -1  1  1 -1 -1  1  1  1
X.8      1  1  1  1 -1 -1 -1 -1  1  1  1
X.9      2  2  .  .  .  .  .  . -2 -2  2
X.10     2 -2  .  .  .  .  .  .  2 -2  2
X.11     4  .  .  .  .  .  .  .  .  . -4