Properties

Label 32T40
Order \(32\)
n \(32\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^2\times Q_8$

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Group action invariants

Degree $n$ :  $32$
Transitive number $t$ :  $40$
Group :  $C_2^2\times Q_8$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,24,2,23)(3,22,4,21)(5,17,6,18)(7,19,8,20)(9,15,10,16)(11,13,12,14)(25,29,26,30)(27,31,28,32), (1,7,2,8)(3,5,4,6)(9,32,10,31)(11,30,12,29)(13,26,14,25)(15,28,16,27)(17,22,18,21)(19,24,20,23), (1,6,2,5)(3,8,4,7)(9,29,10,30)(11,31,12,32)(13,27,14,28)(15,25,16,26)(17,23,18,24)(19,21,20,22), (1,12,2,11)(3,10,4,9)(5,31,6,32)(7,29,8,30)(13,23,14,24)(15,21,16,22)(17,27,18,28)(19,25,20,26)
$|\Aut(F/K)|$:  $32$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 15
4:  $C_2^2$ x 35
8:  $C_2^3$ x 15, $Q_8$ x 4
16:  $C_2^4$, $D_8$ x 6

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 15

Degree 4: $C_2^2$ x 35

Degree 8: $C_2^3$ x 15, $Q_8$ x 4

Degree 16: $C_2^4$, $D_8$ x 6

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 47]
Character table:   
      2  5  5  5  5  4  4  4  4  4  4  4  4  4  4  4  4  5  5  5  5

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

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 -1  1 -1  1 -1  1 -1 -1  1  1 -1 -1  1  1  1 -1 -1
X.10     1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  1  1  1  1  1  1  1  1
X.11     1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1  1  1  1  1
X.12     1  1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1 -1 -1
X.13     1  1  1  1 -1 -1  1  1  1  1 -1 -1  1  1 -1 -1 -1 -1 -1 -1
X.14     1  1  1  1  1  1 -1 -1 -1 -1  1  1  1  1 -1 -1 -1 -1 -1 -1
X.15     1  1  1  1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1 -1 -1
X.16     1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1  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  2 -2  .  .  .  .  .  .  .  .  .  .  .  . -2  2 -2  2
X.20     2 -2  2 -2  .  .  .  .  .  .  .  .  .  .  .  .  2 -2  2 -2