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Properties

Label 16T141
Order \(64\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $D_4:D_4$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $141$
Group :  $D_4:D_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,14)(2,13)(3,4)(5,10)(6,9)(7,15)(8,16)(11,12), (1,3,6,8,10,12,13,15)(2,4,5,7,9,11,14,16), (1,9)(2,10)(3,4)(5,6)(7,15)(8,16)(11,12)(13,14)
$|\Aut(F/K)|$:  $8$

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 6, $C_2^3$
16:  $D_4\times C_2$ x 3
32:  $C_2^2 \wr C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4\times C_2$, $(C_4^2 : C_2):C_2$ x 2

Low degree siblings

8T26 x 4, 16T135 x 2, 16T141, 16T142 x 2, 16T152 x 2, 32T147 x 2, 32T148 x 2, 32T155, 32T156

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 $ $1$ $1$ $()$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $4$ $( 3, 8,12,15)( 4, 7,11,16)$
$ 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 $ $4$ $2$ $( 1, 2)( 3, 4)( 5,13)( 6,14)( 7,15)( 8,16)( 9,10)(11,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 2)( 3, 7)( 4, 8)( 5,13)( 6,14)( 9,10)(11,15)(12,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)$
$ 8, 8 $ $8$ $8$ $( 1, 3, 6, 8,10,12,13,15)( 2, 4, 5, 7, 9,11,14,16)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 3,10,12)( 2, 4, 9,11)( 5, 7,14,16)( 6, 8,13,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 4)( 2, 3)( 5,15)( 6,16)( 7,13)( 8,14)( 9,12)(10,11)$
$ 8, 8 $ $8$ $8$ $( 1, 4, 6,16,10,11,13, 7)( 2, 3, 5,15, 9,12,14, 8)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 4,10,11)( 2, 3, 9,12)( 5,15,14, 8)( 6,16,13, 7)$
$ 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, 2, 2, 2, 2 $ $4$ $4$ $( 1, 6,10,13)( 2, 5, 9,14)( 3,12)( 4,11)( 7,16)( 8,15)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 6,10,13)( 2, 5, 9,14)( 3,15,12, 8)( 4,16,11, 7)$
$ 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:  $64=2^{6}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [64, 134]
Character table:    
      2  6  4  5  4  3  4  3  4  4  3  4  4  5  4  5  6

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

X.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
X.3      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
X.5      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
X.7      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
X.9      2  .  2 -2  .  .  .  .  .  .  .  2 -2  . -2  2
X.10     2  .  2  2  .  .  .  .  .  .  . -2 -2  . -2  2
X.11     2  . -2  .  . -2  .  2  .  .  .  . -2  .  2  2
X.12     2  . -2  .  .  .  .  . -2  .  2  .  2  . -2  2
X.13     2  . -2  .  .  .  .  .  2  . -2  .  2  . -2  2
X.14     2  . -2  .  .  2  . -2  .  .  .  . -2  .  2  2
X.15     4 -2  .  .  .  .  .  .  .  .  .  .  .  2  . -4
X.16     4  2  .  .  .  .  .  .  .  .  .  .  . -2  . -4