Properties

Label 32T35
Order \(32\)
n \(32\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2\times C_4:C_4$

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Group action invariants

Degree $n$ :  $32$
Transitive number $t$ :  $35$
Group :  $C_2\times C_4:C_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (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), (1,14,12,23)(2,13,11,24)(3,16,9,22)(4,15,10,21)(5,20,30,28)(6,19,29,27)(7,18,31,26)(8,17,32,25), (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)
$|\Aut(F/K)|$:  $32$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_4$ x 4, $C_2^2$ x 7
8:  $D_{4}$ x 2, $C_4\times C_2$ x 6, $C_2^3$, $Q_8$ x 2
16:  $D_4\times C_2$, $C_4\times C_2^2$, $D_8$, $C_4:C_4$ x 4

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_4$ x 4, $C_2^2$ x 7, $D_{4}$ x 4

Degree 8: $C_4\times C_2$ x 6, $C_2^3$, $D_4$ x 2, $Q_8$ x 2, $D_4\times C_2$ x 4

Degree 16: $C_4\times C_2^2$, $D_8$, $C_4:C_4$ x 4, $D_4\times C_2$

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 23]
Character table:   
      2  5  5  4  4  4  4  5  5  4  4  4  5  5  4  4  4  5  5  4  4

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

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  A -A  1 -1 -1  A -A  1 -1 -1  A -A -1  1  1 -A  A
X.10     1  1 -1 -A  A  1 -1 -1 -A  A  1 -1 -1 -A  A -1  1  1  A -A
X.11     1  1 -1  A -A  1 -1 -1 -A  A -1  1  1 -A  A  1 -1 -1 -A  A
X.12     1  1 -1 -A  A  1 -1 -1  A -A -1  1  1  A -A  1 -1 -1  A -A
X.13     1  1  1  A  A -1 -1 -1  A  A -1 -1 -1 -A -A  1  1  1 -A -A
X.14     1  1  1 -A -A -1 -1 -1 -A -A -1 -1 -1  A  A  1  1  1  A  A
X.15     1  1  1  A  A -1 -1 -1 -A -A  1  1  1  A  A -1 -1 -1 -A -A
X.16     1  1  1 -A -A -1 -1 -1  A  A  1  1  1 -A -A -1 -1 -1  A  A
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  .  .

A = -E(4)
  = -Sqrt(-1) = -i