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Properties

Label 16T396
Order \(128\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $D_4^2.C_2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $396$
Group :  $D_4^2.C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $4$
Generators:  (7,15)(8,16), (1,5)(2,6)(3,4)(7,8)(9,13)(10,14)(11,12)(15,16), (1,4,6,15)(2,3,5,16)(7,9,12,14)(8,10,11,13)
$|\Aut(F/K)|$:  $4$

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:  $C_2^2 \wr C_2$
64:  $(((C_4 \times C_2): C_2):C_2):C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4$, $C_2 \wr C_2\wr C_2$ x 2

Low degree siblings

8T35 x 8, 16T376 x 4, 16T388 x 4, 16T390 x 4, 16T391 x 4, 16T393 x 4, 16T395 x 4, 16T396 x 3, 16T401 x 4, 32T852 x 4, 32T853 x 2, 32T854 x 2, 32T872 x 2, 32T876 x 4, 32T877 x 2, 32T880 x 2, 32T882 x 2, 32T883 x 4, 32T884 x 2, 32T885 x 2

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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 7,15)( 8,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 5,13)( 6,14)( 7,15)( 8,16)$
$ 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, 1, 1, 1, 1 $ $4$ $2$ $( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 2)( 3, 7)( 4, 8)( 5, 6)( 9,10)(11,15)(12,16)(13,14)$
$ 4, 4, 2, 2, 2, 2 $ $4$ $4$ $( 1, 2)( 3, 7,11,15)( 4, 8,12,16)( 5, 6)( 9,10)(13,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 2)( 3, 7)( 4, 8)( 5,14)( 6,13)( 9,10)(11,15)(12,16)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 2)( 3, 7,11,15)( 4, 8,12,16)( 5,14)( 6,13)( 9,10)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1, 3)( 2, 4)( 5, 7,13,15)( 6, 8,14,16)( 9,11)(10,12)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 3, 9,11)( 2, 4,10,12)( 5, 7,13,15)( 6, 8,14,16)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 4, 6, 7)( 2, 3, 5, 8)( 9,12,14,15)(10,11,13,16)$
$ 8, 8 $ $16$ $8$ $( 1, 4, 6, 7, 9,12,14,15)( 2, 3, 5, 8,10,11,13,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 5)( 2, 6)( 3,12)( 4,11)( 7,16)( 8,15)( 9,13)(10,14)$
$ 4, 4, 2, 2, 2, 2 $ $4$ $4$ $( 1, 5, 9,13)( 2, 6,10,14)( 3,12)( 4,11)( 7,16)( 8,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,14)(10,13)(11,16)(12,15)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 6)( 2, 5)( 3, 8,11,16)( 4, 7,12,15)( 9,14)(10,13)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 6, 9,14)( 2, 5,10,13)( 3, 8,11,16)( 4, 7,12,15)$
$ 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:  $128=2^{7}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [128, 928]
Character table:    
      2  7  5  5  6  5  5  5  4  4  4  3  4  3  3  5  5  5  4  5  7

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

X.1      1  1  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 -1  1  1
X.3      1 -1  1  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 -1  1  1
X.5      1 -1  1  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  1  1  1
X.7      1  1  1  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  1  1  1
X.9      2  2  2  2  2  .  .  .  .  .  .  .  .  .  .  . -2 -2 -2  2
X.10     2 -2  2  2 -2  .  .  .  .  .  .  .  .  .  .  . -2  2 -2  2
X.11     2  . -2  2  .  2  .  . -2  .  .  .  .  .  2  .  2  . -2  2
X.12     2  . -2  2  . -2  .  .  2  .  .  .  .  . -2  .  2  . -2  2
X.13     2  . -2  2  .  . -2  2  .  .  .  .  .  .  . -2 -2  .  2  2
X.14     2  . -2  2  .  .  2 -2  .  .  .  .  .  .  .  2 -2  .  2  2
X.15     4  .  . -4  .  .  .  .  . -2  .  2  .  .  .  .  .  .  .  4
X.16     4  .  . -4  .  .  .  .  .  2  . -2  .  .  .  .  .  .  .  4
X.17     4 -2  .  .  2 -2  2  .  .  .  .  .  .  .  2 -2  .  .  . -4
X.18     4 -2  .  .  2  2 -2  .  .  .  .  .  .  . -2  2  .  .  . -4
X.19     4  2  .  . -2 -2 -2  .  .  .  .  .  .  .  2  2  .  .  . -4
X.20     4  2  .  . -2  2  2  .  .  .  .  .  .  . -2 -2  .  .  . -4