Properties

Label 32T11
Order \(32\)
n \(32\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^2\times D_4$

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Group action invariants

Degree $n$ :  $32$
Transitive number $t$ :  $11$
Group :  $C_2^2\times D_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,18)(2,17)(3,19)(4,20)(5,24)(6,23)(7,22)(8,21)(9,27)(10,28)(11,26)(12,25)(13,32)(14,31)(15,29)(16,30), (1,7)(2,8)(3,6)(4,5)(9,30)(10,29)(11,32)(12,31)(13,23)(14,24)(15,21)(16,22)(17,28)(18,27)(19,26)(20,25), (1,23,30,26)(2,24,29,25)(3,22,32,27)(4,21,31,28)(5,17,12,15)(6,18,11,16)(7,19,9,13)(8,20,10,14), (1,10)(2,9)(3,12)(4,11)(5,32)(6,31)(7,29)(8,30)(13,24)(14,23)(15,22)(16,21)(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 15
4:  $C_2^2$ x 35
8:  $D_{4}$ x 4, $C_2^3$ x 15
16:  $D_4\times C_2$ x 6, $C_2^4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 15

Degree 4: $C_2^2$ x 35, $D_{4}$ x 8

Degree 8: $C_2^3$ x 15, $D_4$ x 4, $D_4\times C_2$ x 24

Degree 16: $C_2^4$, $D_4\times C_2$ x 6, $C_2^2 \times D_4$ x 8

Low degree siblings

16T25 x 8

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 46]
Character table:   
      2  5  4  5  4  5  4  4  5  5  5  4  4  4  4  4  4  4  4  5  5

        1a 2a 2b 2c 2d 2e 2f 2g 2h 2i 2j 4a 4b 2k 2l 4c 4d 2m 2n 2o
     2P 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 1a 2n 2n 1a 1a 2n 2n 1a 1a 1a
     3P 1a 2a 2b 2c 2d 2e 2f 2g 2h 2i 2j 4a 4b 2k 2l 4c 4d 2m 2n 2o

X.1      1  1  1  1  1  1  1  1  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  1 -1  1  1 -1  1 -1  1 -1
X.3      1 -1 -1  1 -1  1 -1  1  1 -1  1 -1  1 -1 -1  1 -1  1  1 -1
X.4      1 -1 -1  1  1 -1  1 -1 -1  1 -1  1 -1  1 -1  1 -1  1  1 -1
X.5      1 -1 -1  1  1 -1  1 -1 -1  1  1 -1  1 -1  1 -1  1 -1  1 -1
X.6      1 -1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1  1 -1 -1  1  1  1
X.7      1 -1  1 -1 -1  1  1 -1 -1 -1  1 -1 -1  1 -1  1  1 -1  1  1
X.8      1 -1  1 -1  1 -1 -1  1  1  1 -1  1  1 -1 -1  1  1 -1  1  1
X.9      1 -1  1 -1  1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1  1  1  1
X.10     1  1 -1 -1 -1 -1  1  1  1 -1 -1 -1  1  1  1  1 -1 -1  1 -1
X.11     1  1 -1 -1 -1 -1  1  1  1 -1  1  1 -1 -1 -1 -1  1  1  1 -1
X.12     1  1 -1 -1  1  1 -1 -1 -1  1 -1 -1  1  1 -1 -1  1  1  1 -1
X.13     1  1 -1 -1  1  1 -1 -1 -1  1  1  1 -1 -1  1  1 -1 -1  1 -1
X.14     1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1
X.15     1  1  1  1 -1 -1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1
X.16     1  1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1
X.17     2  . -2  . -2  .  .  2 -2  2  .  .  .  .  .  .  .  . -2  2
X.18     2  . -2  .  2  .  . -2  2 -2  .  .  .  .  .  .  .  . -2  2
X.19     2  .  2  . -2  .  . -2  2  2  .  .  .  .  .  .  .  . -2 -2
X.20     2  .  2  .  2  .  .  2 -2 -2  .  .  .  .  .  .  .  . -2 -2