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Properties

Label 16T10
Order \(16\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^2 : C_4$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $10$
Group :  $C_2^2 : C_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,9)(2,10)(3,4)(5,13)(6,14)(7,8)(11,12)(15,16), (1,3,6,15)(2,4,5,16)(7,9,11,14)(8,10,12,13)
$|\Aut(F/K)|$:  $16$

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, $C_4\times C_2$

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: $C_4\times C_2$, $D_4$ x 2, $C_2^2:C_4$ x 2

Low degree siblings

8T10 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 $ $1$ $1$ $()$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 2)( 3,11)( 4,12)( 5, 6)( 7,15)( 8,16)( 9,10)(13,14)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 3, 6,15)( 2, 4, 5,16)( 7, 9,11,14)( 8,10,12,13)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 4,13, 7)( 2, 3,14, 8)( 5,15, 9,12)( 6,16,10,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,13)(10,14)(11,15)(12,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 6)( 2, 5)( 3,15)( 4,16)( 7,11)( 8,12)( 9,14)(10,13)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 7,13, 4)( 2, 8,14, 3)( 5,12, 9,15)( 6,11,10,16)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 8, 6,12)( 2, 7, 5,11)( 3,10,15,13)( 4, 9,16,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,13)( 2,14)( 3, 8)( 4, 7)( 5, 9)( 6,10)(11,16)(12,15)$

Group invariants

Order:  $16=2^{4}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [16, 3]
Character table:    
      2  4  3  3  3  3  4  3  3  4  4

        1a 2a 4a 4b 2b 2c 4c 4d 2d 2e
     2P 1a 1a 2c 2e 1a 1a 2e 2c 1a 1a
     3P 1a 2a 4d 4c 2b 2c 4b 4a 2d 2e

X.1      1  1  1  1  1  1  1  1  1  1
X.2      1 -1 -1  1 -1  1  1 -1  1  1
X.3      1 -1  1 -1 -1  1 -1  1  1  1
X.4      1  1 -1 -1  1  1 -1 -1  1  1
X.5      1 -1  A -A  1 -1  A -A  1 -1
X.6      1 -1 -A  A  1 -1 -A  A  1 -1
X.7      1  1  A  A -1 -1 -A -A  1 -1
X.8      1  1 -A -A -1 -1  A  A  1 -1
X.9      2  .  .  .  .  2  .  . -2 -2
X.10     2  .  .  .  . -2  .  . -2  2

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