Properties

Label 16T49
Order \(32\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_8.C_4$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $49$
Group :  $C_8.C_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,4,6,8,2,3,5,7)(9,12,14,16,10,11,13,15), (1,12,5,16,2,11,6,15)(3,10,8,13,4,9,7,14)
$|\Aut(F/K)|$:  $8$

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:  $D_{4}$, $C_4\times C_2$, $Q_8$
16:  $C_4:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 8: $Q_8$

Low degree siblings

32T28

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 15]
Character table:   
      2  5  4  5  4  4  4  4  4  5  5  3  3  3  3

        1a 2a 2b 8a 8b 8c 8d 4a 4b 4c 8e 8f 8g 8h
     2P 1a 1a 1a 4a 4a 4a 4a 2b 2b 2b 4b 4c 4c 4b
     3P 1a 2a 2b 8d 8b 8c 8a 4a 4c 4b 8f 8e 8h 8g
     5P 1a 2a 2b 8d 8c 8b 8a 4a 4b 4c 8e 8f 8g 8h
     7P 1a 2a 2b 8a 8c 8b 8d 4a 4c 4b 8f 8e 8h 8g

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  D -D  D -D
X.6      1 -1  1 -1  1  1 -1  1 -1 -1 -D  D -D  D
X.7      1 -1  1  1 -1 -1  1  1 -1 -1  D -D -D  D
X.8      1 -1  1  1 -1 -1  1  1 -1 -1 -D  D  D -D
X.9      2  2  2  .  .  .  . -2 -2 -2  .  .  .  .
X.10     2 -2  2  .  .  .  . -2  2  2  .  .  .  .
X.11     2  . -2  A  B -B -A  .  C -C  .  .  .  .
X.12     2  . -2  A -B  B -A  . -C  C  .  .  .  .
X.13     2  . -2 -A  B -B  A  . -C  C  .  .  .  .
X.14     2  . -2 -A -B  B  A  .  C -C  .  .  .  .

A = -E(8)+E(8)^3
  = -Sqrt(2) = -r2
B = -E(8)-E(8)^3
  = -Sqrt(-2) = -i2
C = -2*E(4)
  = -2*Sqrt(-1) = -2i
D = -E(4)
  = -Sqrt(-1) = -i