Properties

Label 32T10
Order \(32\)
n \(32\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^2:C_8$

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Group action invariants

Degree $n$ :  $32$
Transitive number $t$ :  $10$
Group :  $C_2^2:C_8$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,22,10,29,19,5,27,13)(2,21,9,30,20,6,28,14)(3,24,11,31,18,8,26,16)(4,23,12,32,17,7,25,15), (1,9,19,28)(2,10,20,27)(3,12,18,25)(4,11,17,26)(5,32,22,15)(6,31,21,16)(7,29,23,13)(8,30,24,14)
$|\Aut(F/K)|$:  $32$

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:  $D_{4}$ x 2, $C_8$ x 2, $C_4\times C_2$
16:  $C_8:C_2$, $C_2^2:C_4$, $C_8\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $C_8$ x 2, $C_4\times C_2$, $D_4$ x 2, $C_8:C_2$, $C_2^2:C_4$ x 2

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

Low degree siblings

16T24 x 2

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

Group invariants

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

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

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   A  -A   A  -A  B -B  B -B -/A  /A -/A  /A  1 -1 -B  B
X.6      1 -1 -1  1 -/A  /A -/A  /A -B  B -B  B   A  -A   A  -A  1 -1  B -B
X.7      1 -1 -1  1  /A -/A  /A -/A -B  B -B  B  -A   A  -A   A  1 -1  B -B
X.8      1 -1 -1  1  -A   A  -A   A  B -B  B -B  /A -/A  /A -/A  1 -1 -B  B
X.9      1 -1  1 -1   B  -B  -B   B  1 -1 -1  1  -B   B   B  -B  1  1 -1 -1
X.10     1 -1  1 -1  -B   B   B  -B  1 -1 -1  1   B  -B  -B   B  1  1 -1 -1
X.11     1  1 -1 -1   A   A  -A  -A -B -B  B  B -/A -/A  /A  /A  1 -1 -B  B
X.12     1  1 -1 -1 -/A -/A  /A  /A  B  B -B -B   A   A  -A  -A  1 -1  B -B
X.13     1  1 -1 -1  /A  /A -/A -/A  B  B -B -B  -A  -A   A   A  1 -1  B -B
X.14     1  1 -1 -1  -A  -A   A   A -B -B  B  B  /A  /A -/A -/A  1 -1 -B  B
X.15     1  1  1  1   B   B   B   B -1 -1 -1 -1  -B  -B  -B  -B  1  1 -1 -1
X.16     1  1  1  1  -B  -B  -B  -B -1 -1 -1 -1   B   B   B   B  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  .   .   .   .   .  .  C  C  .   .   .   .   . -2 -2 -C -C
X.20     2  .  2  .   .   .   .   .  . -C -C  .   .   .   .   . -2 -2  C  C

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