Properties

Label 32T8
Order \(32\)
n \(32\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $OD_{32}$

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Group action invariants

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

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:  $C_8$ x 2, $C_4\times C_2$
16:  $C_8\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

Degree 8: $C_8$ x 2, $C_4\times C_2$

Degree 16: $C_8\times C_2$, $C_{16} : C_2$

Low degree siblings

16T22

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 17]
Character table:   
      2  5  4  5  4   5   5  5  4  5   5  4   5   4   4   4   4   4   4   4

        1a 2a 2b 8a  8b  8c 4a 4b 4c  8d 8e  8f 16a 16b 16c 16d 16e 16f 16g
     2P 1a 1a 1a 4c  4c  4c 2b 2b 2b  4a 4a  4a  8c  8b  8f  8d  8b  8c  8d
     3P 1a 2a 2b 8e  8d  8f 4c 4b 4a  8b 8a  8c 16c 16d 16a 16b 16g 16h 16e
     5P 1a 2a 2b 8a  8c  8b 4a 4b 4c  8f 8e  8d 16e 16f 16g 16h 16a 16b 16c
     7P 1a 2a 2b 8e  8f  8d 4c 4b 4a  8c 8a  8b 16g 16h 16e 16f 16c 16d 16a
    11P 1a 2a 2b 8e  8d  8f 4c 4b 4a  8b 8a  8c 16c 16d 16a 16b 16g 16h 16e
    13P 1a 2a 2b 8a  8c  8b 4a 4b 4c  8f 8e  8d 16e 16f 16g 16h 16a 16b 16c

X.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
X.3      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
X.5      1 -1  1  1  -1  -1  1 -1  1  -1  1  -1   A  -A  -A   A   A  -A  -A
X.6      1 -1  1  1  -1  -1  1 -1  1  -1  1  -1  -A   A   A  -A  -A   A   A
X.7      1 -1  1  A  -A  -A -1  1 -1   A -A   A   D  -D -/D  /D  -D   D  /D
X.8      1 -1  1  A  -A  -A -1  1 -1   A -A   A  -D   D  /D -/D   D  -D -/D
X.9      1 -1  1 -A   A   A -1  1 -1  -A  A  -A -/D  /D   D  -D  /D -/D  -D
X.10     1 -1  1 -A   A   A -1  1 -1  -A  A  -A  /D -/D  -D   D -/D  /D   D
X.11     1  1  1 -1  -1  -1  1  1  1  -1 -1  -1   A   A  -A  -A   A   A  -A
X.12     1  1  1 -1  -1  -1  1  1  1  -1 -1  -1  -A  -A   A   A  -A  -A   A
X.13     1  1  1  A   A   A -1 -1 -1  -A -A  -A -/D -/D   D   D  /D  /D  -D
X.14     1  1  1  A   A   A -1 -1 -1  -A -A  -A  /D  /D  -D  -D -/D -/D   D
X.15     1  1  1 -A  -A  -A -1 -1 -1   A  A   A   D   D -/D -/D  -D  -D  /D
X.16     1  1  1 -A  -A  -A -1 -1 -1   A  A   A  -D  -D  /D  /D   D   D -/D
X.17     2  . -2  .   B  -B  C  . -C -/B  .  /B   .   .   .   .   .   .   .
X.18     2  . -2  . -/B  /B -C  .  C   B  .  -B   .   .   .   .   .   .   .
X.19     2  . -2  .  /B -/B -C  .  C  -B  .   B   .   .   .   .   .   .   .
X.20     2  . -2  .  -B   B  C  . -C  /B  . -/B   .   .   .   .   .   .   .

      2   4

        16h
     2P  8f
     3P 16f
     5P 16d
     7P 16b
    11P 16f
    13P 16d

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

A = -E(4)
  = -Sqrt(-1) = -i
B = -2*E(8)
C = -2*E(4)
  = -2*Sqrt(-1) = -2i
D = -E(8)