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Properties

Label 32T30
Order \(32\)
n \(32\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $SD_{32}$

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Group action invariants

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

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4$, $D_{8}$ x 2

Degree 16: $D_{8}$, 16T55

Low degree siblings

16T55

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 19]
Character table:    
      2  5  2  5   4  2   4  4  4   4   4  4

        1a 2a 2b 16a 4a 16b 8a 8b 16c 16d 4b
     2P 1a 1a 1a  8a 2b  8a 4b 4b  8b  8b 2b
     3P 1a 2a 2b 16c 4a 16d 8b 8a 16b 16a 4b
     5P 1a 2a 2b 16c 4a 16d 8b 8a 16b 16a 4b
     7P 1a 2a 2b 16a 4a 16b 8a 8b 16c 16d 4b
    11P 1a 2a 2b 16d 4a 16c 8b 8a 16a 16b 4b
    13P 1a 2a 2b 16d 4a 16c 8b 8a 16a 16b 4b

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      2  .  2   .  .   . -2 -2   .   .  2
X.6      2  .  2   A  .   A  .  .  -A  -A -2
X.7      2  .  2  -A  .  -A  .  .   A   A -2
X.8      2  . -2   B  .  -B -A  A   C  -C  .
X.9      2  . -2   C  .  -C  A -A  -B   B  .
X.10     2  . -2  -C  .   C  A -A   B  -B  .
X.11     2  . -2  -B  .   B -A  A  -C   C  .

A = -E(8)+E(8)^3
  = -Sqrt(2) = -r2
B = -E(16)-E(16)^7
C = -E(16)^3-E(16)^5