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Properties

Label 16T31
Order \(32\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^2:Q_8$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $31$
Group :  $C_2^2:Q_8$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,8,16,10)(2,7,15,9)(3,14,5,12)(4,13,6,11), (1,6,2,5)(3,16,4,15)(7,11,8,12)(9,13,10,14), (1,2)(3,4)(5,6)(7,9)(8,10)(11,13)(12,14)(15,16)
$|\Aut(F/K)|$:  $8$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $D_{4}$ x 2, $C_2^3$, $Q_8$ x 2
16:  $D_4\times C_2$, $Q_8:C_2$, $D_8$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $Q_8$, $D_4\times C_2$, $Q_8:C_2$

Low degree siblings

16T31, 32T17

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 29]
Character table:    
      2  5  4  5  4  4  4  4  4  3  3  3  3  5  5

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

X.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
X.3      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
X.5      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
X.7      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
X.9      2  2 -2 -2  .  .  .  .  .  .  .  .  2 -2
X.10     2 -2 -2  2  .  .  .  .  .  .  .  .  2 -2
X.11     2  .  2  .  . -2  .  2  .  .  .  . -2 -2
X.12     2  .  2  .  .  2  . -2  .  .  .  . -2 -2
X.13     2  . -2  .  A  . -A  .  .  .  .  . -2  2
X.14     2  . -2  . -A  .  A  .  .  .  .  . -2  2

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