Properties

Label 16T24
Order \(32\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^2 : C_8$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $24$
Group :  $C_2^2 : C_8$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,3,5,7,10,11,14,15)(2,4,6,8,9,12,13,16), (1,10)(2,9)(3,4)(5,14)(6,13)(7,8)(11,12)(15,16)
$|\Aut(F/K)|$:  $8$

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:  $D_{4}$ x 2, $C_8$ x 2, $C_4\times C_2$
16:  $C_8:C_2$, $C_2^2:C_4$, $C_8\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$, $D_{4}$ x 2

Degree 8: $C_8$, $C_8:C_2$, $C_2^2:C_4$

Low degree siblings

16T24, 32T10

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

Group invariants

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

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

X.1      1  1  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  1  1  1
X.3      1 -1  1 -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  1  1  1
X.5      1 -1 -1  1   A  -A  -A   A -B  B  B -B  /A -/A -/A  /A  1 -1 -B  B
X.6      1 -1 -1  1 -/A  /A  /A -/A  B -B -B  B  -A   A   A  -A  1 -1  B -B
X.7      1 -1 -1  1  /A -/A -/A  /A  B -B -B  B   A  -A  -A   A  1 -1  B -B
X.8      1 -1 -1  1  -A   A   A  -A -B  B  B -B -/A  /A  /A -/A  1 -1 -B  B
X.9      1 -1  1 -1   B  -B   B  -B -1  1 -1  1   B  -B   B  -B  1  1 -1 -1
X.10     1 -1  1 -1  -B   B  -B   B -1  1 -1  1  -B   B  -B   B  1  1 -1 -1
X.11     1  1 -1 -1   A   A  -A  -A -B -B  B  B -/A -/A  /A  /A  1 -1 -B  B
X.12     1  1 -1 -1 -/A -/A  /A  /A  B  B -B -B   A   A  -A  -A  1 -1  B -B
X.13     1  1 -1 -1  /A  /A -/A -/A  B  B -B -B  -A  -A   A   A  1 -1  B -B
X.14     1  1 -1 -1  -A  -A   A   A -B -B  B  B  /A  /A -/A -/A  1 -1 -B  B
X.15     1  1  1  1   B   B   B   B -1 -1 -1 -1  -B  -B  -B  -B  1  1 -1 -1
X.16     1  1  1  1  -B  -B  -B  -B -1 -1 -1 -1   B   B   B   B  1  1 -1 -1
X.17     2  . -2  .   .   .   .   . -2  .  2  .   .   .   .   . -2  2  2 -2
X.18     2  . -2  .   .   .   .   .  2  . -2  .   .   .   .   . -2  2 -2  2
X.19     2  .  2  .   .   .   .   .  C  .  C  .   .   .   .   . -2 -2 -C -C
X.20     2  .  2  .   .   .   .   . -C  . -C  .   .   .   .   . -2 -2  C  C

A = -E(8)
B = -E(4)
  = -Sqrt(-1) = -i
C = -2*E(4)
  = -2*Sqrt(-1) = -2i