Properties

Label 32T48
Order \(32\)
n \(32\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2.D_8$

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Group action invariants

Degree $n$ :  $32$
Transitive number $t$ :  $48$
Group :  $C_2.D_8$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,30,11,6)(2,29,12,5)(3,32,9,8)(4,31,10,7)(13,28,21,18)(14,27,22,17)(15,25,23,19)(16,26,24,20), (1,18,3,20,2,17,4,19)(5,24,8,21,6,23,7,22)(9,26,12,27,10,25,11,28)(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 2, $C_2^2$
8:  $D_{4}$, $C_4\times C_2$, $Q_8$
16:  $D_{8}$, $C_4:C_4$, $Q_{16}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $C_4\times C_2$, $D_4$, $Q_8$, $D_{8}$ x 2

Degree 16: $C_4:C_4$, $D_{8}$, $Q_{16}$

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 14]
Character table:   
      2  5  5  4  3  4  5  5  3  4  4  3  4  4  3

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

X.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
X.3      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
X.5      1  1  1  A -1 -1 -1  A -1 -1 -A  1  1 -A
X.6      1  1  1 -A -1 -1 -1 -A -1 -1  A  1  1  A
X.7      1  1  1  A -1 -1 -1 -A  1  1  A -1 -1 -A
X.8      1  1  1 -A -1 -1 -1  A  1  1 -A -1 -1  A
X.9      2  2 -2  . -2  2  2  .  .  .  .  .  .  .
X.10     2  2 -2  .  2 -2 -2  .  .  .  .  .  .  .
X.11     2 -2  .  .  . -2  2  .  B -B  .  B -B  .
X.12     2 -2  .  .  . -2  2  . -B  B  . -B  B  .
X.13     2 -2  .  .  .  2 -2  .  B -B  . -B  B  .
X.14     2 -2  .  .  .  2 -2  . -B  B  .  B -B  .

A = -E(4)
  = -Sqrt(-1) = -i
B = -E(8)+E(8)^3
  = -Sqrt(2) = -r2