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Properties

Label 32T9
Order \(32\)
n \(32\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $Q_8:C_2^2$

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Group action invariants

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

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
16:  $C_2^4$

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, $C_2^3 : D_4 $ x 6

Degree 16: $C_2^4$, $Q_8 : C_2^2$ x 9

Low degree siblings

8T22 x 6, 16T23 x 9

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 49]
Character table:    
      2  5  4  4  4  4  4  4  4  4  4  4  4  4  4  4  4  5

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

X.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
X.3      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
X.5      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
X.7      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
X.9      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
X.11     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
X.13     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
X.15     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
X.17     4  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . -4