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Properties

Label 16T36
Order \(32\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^3.C_4$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $36$
Group :  $C_2^3.C_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,3,6,8,10,12,13,15)(2,4,5,7,9,11,14,16), (1,2)(3,11)(4,12)(5,13)(6,14)(7,8)(9,10)(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, $C_4\times C_2$
16:  $C_2^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

Degree 8: $C_4\times C_2$, $(C_8:C_2):C_2$ x 2

Low degree siblings

8T16 x 2, 16T41 x 2, 32T22

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 7]
Character table:    
      2  5  4  3  3  3  3  4  4  3  3  5

        1a 2a 2b 8a 8b 2c 4a 4b 8c 8d 2d
     2P 1a 1a 1a 4a 4b 1a 2d 2d 4b 4a 1a
     3P 1a 2a 2b 8d 8c 2c 4a 4b 8b 8a 2d
     5P 1a 2a 2b 8a 8b 2c 4a 4b 8c 8d 2d
     7P 1a 2a 2b 8d 8c 2c 4a 4b 8b 8a 2d

X.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
X.3      1  1 -1  1 -1 -1  1  1 -1  1  1
X.4      1  1  1 -1 -1  1  1  1 -1 -1  1
X.5      1  1 -1  A -A  1 -1 -1  A -A  1
X.6      1  1 -1 -A  A  1 -1 -1 -A  A  1
X.7      1  1  1  A  A -1 -1 -1 -A -A  1
X.8      1  1  1 -A -A -1 -1 -1  A  A  1
X.9      2 -2  .  .  .  . -2  2  .  .  2
X.10     2 -2  .  .  .  .  2 -2  .  .  2
X.11     4  .  .  .  .  .  .  .  .  . -4

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