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Properties

Label 16T20
Order \(32\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $(C_2 \times Q_8):C_2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $20$
Group :  $(C_2 \times Q_8):C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,7,2,8)(3,6,4,5)(9,12,10,11)(13,15,14,16), (1,2)(3,4)(5,6)(7,8), (1,4,2,3)(5,7,6,8)(9,16,10,15)(11,13,12,14), (1,16)(2,15)(3,9)(4,10)(5,11)(6,12)(7,13)(8,14)
$|\Aut(F/K)|$:  $8$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 15
4:  $C_2^2$ x 35
8:  $C_2^3$ x 15
16:  $C_2^4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7

Degree 8: $C_2^3$

Low degree siblings

16T20 x 4, 32T6

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

Group invariants

Order:  $32=2^{5}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [32, 50]
Character table:    
      2  5  4  5  4  4  4  4  4  4  4  4  4  4  4  4  4  4

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

X.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
X.3      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
X.5      1 -1  1 -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 -1 -1  1
X.7      1 -1  1  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  1 -1  1
X.9      1 -1  1  1 -1  1 -1 -1  1  1 -1 -1  1  1 -1  1 -1
X.10     1  1  1 -1 -1 -1 -1  1  1 -1 -1 -1 -1  1  1  1  1
X.11     1  1  1 -1 -1 -1 -1  1  1  1  1  1  1 -1 -1 -1 -1
X.12     1  1  1 -1 -1  1  1 -1 -1 -1 -1  1  1 -1 -1  1  1
X.13     1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1  1  1 -1 -1
X.14     1  1  1  1  1 -1 -1 -1 -1 -1 -1  1  1  1  1 -1 -1
X.15     1  1  1  1  1 -1 -1 -1 -1  1  1 -1 -1 -1 -1  1  1
X.16     1  1  1  1  1  1  1  1  1 -1 -1 -1 -1 -1 -1 -1 -1
X.17     4  . -4  .  .  .  .  .  .  .  .  .  .  .  .  .  .