Properties

Label 16T23
Order \(32\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $Q_8 : C_2^2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $23$
Group :  $Q_8 : C_2^2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,16)(2,15)(3,6)(4,5)(7,10)(8,9)(11,14)(12,13), (1,5)(2,6)(3,15)(4,16)(7,12)(8,11)(9,14)(10,13), (3,6)(4,5)(11,14)(12,13), (1,2)(3,5)(4,6)(7,9)(8,10)(11,12)(13,14)(15,16), (1,10)(2,9)(3,11)(4,12)(5,13)(6,14)(7,16)(8,15)
$|\Aut(F/K)|$:  $8$

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 7

Degree 4: $C_2^2$ x 7

Degree 8: $C_2^3$, $C_2^3 : D_4 $ x 2

Low degree siblings

8T22 x 6, 16T23 x 8, 32T9

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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 3, 6)( 4, 5)(11,14)(12,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 9)( 8,10)(11,13)(12,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 2)( 3, 5)( 4, 6)( 7, 9)( 8,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 3)( 2, 4)( 5,15)( 6,16)( 7,11)( 8,12)( 9,13)(10,14)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 3,16, 6)( 2, 4,15, 5)( 7,11,10,14)( 8,12, 9,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 4)( 2, 3)( 5,16)( 6,15)( 7,13)( 8,14)( 9,11)(10,12)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 4,16, 5)( 2, 3,15, 6)( 7,13,10,12)( 8,14, 9,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 7)( 2, 8)( 3,11)( 4,12)( 5,13)( 6,14)( 9,15)(10,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 7)( 2, 8)( 3,14)( 4,13)( 5,12)( 6,11)( 9,15)(10,16)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 8,16, 9)( 2, 7,15,10)( 3,12, 6,13)( 4,11, 5,14)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 8,16, 9)( 2, 7,15,10)( 3,13, 6,12)( 4,14, 5,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1,11)( 2,12)( 3, 7)( 4, 8)( 5, 9)( 6,10)(13,15)(14,16)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1,11,16,14)( 2,12,15,13)( 3,10, 6, 7)( 4, 9, 5, 8)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1,12,16,13)( 2,11,15,14)( 3, 8, 6, 9)( 4, 7, 5,10)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1,12)( 2,11)( 3, 9)( 4,10)( 5, 7)( 6, 8)(13,16)(14,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,16)( 2,15)( 3, 6)( 4, 5)( 7,10)( 8, 9)(11,14)(12,13)$

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