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Properties

Label 16T54
Order \(32\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $(C_2^2\times C_4):C_2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $54$
Group :  $(C_2^2\times C_4):C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,15,10,7)(2,16,9,8)(3,13,12,6)(4,14,11,5), (1,7,2,8)(3,14,4,13)(5,11,6,12)(9,16,10,15), (1,4)(2,3)(5,8)(6,7)(9,12)(10,11)(13,15)(14,16)
$|\Aut(F/K)|$:  $8$

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, $C_2^3$
16:  $D_4\times C_2$, $Q_8:C_2$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4$, $Q_8:C_2$ x 2

Low degree siblings

16T37, 16T54, 32T23

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 30]
Character table:    
      2  5  4  5  4  3  3  4  4  4  3  3  5  5  4

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

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  .  .  .  .  A -A  .  . -2  2  .
X.12     2  . -2  .  .  .  . -A  A  .  . -2  2  .
X.13     2  .  2  .  .  .  A  .  .  .  . -2 -2 -A
X.14     2  .  2  .  .  . -A  .  .  .  . -2 -2  A

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