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Properties

Label 16T35
Order \(32\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_8:C_2^2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $35$
Group :  $C_8:C_2^2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,11)(2,12)(3,9)(4,10)(5,7)(6,8)(13,15)(14,16), (1,4,6,7,9,12,14,15)(2,3,5,8,10,11,13,16), (1,9)(2,10)(5,13)(6,14)
$|\Aut(F/K)|$:  $8$

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:  $D_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4$, $Z_8 : Z_8^\times$ x 2

Low degree siblings

8T15 x 2, 16T38 x 2, 16T45, 32T21

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 43]
Character table:    
      2  5  4  3  3  3  3  3  3  4  4  5

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

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