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Properties

Label 16T408
Order \(128\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^3.(C_2\times D_4)$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $408$
Group :  $C_2^3.(C_2\times D_4)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,4,13,15,10,11,6,8)(2,3,14,16,9,12,5,7), (1,5,10,14)(2,6,9,13)(3,4)(7,8)(11,12)(15,16), (1,12,2,11)(3,9,4,10)(5,8,6,7)(13,16,14,15)
$|\Aut(F/K)|$:  $4$

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 12, $C_2^3$
16:  $D_4\times C_2$ x 6, $Q_8:C_2$
32:  $C_2^2 \wr C_2$ x 3, 16T34 x 3, $C_4^2:C_2$
64:  $(C_4^2 : C_2):C_2$ x 2, 32T320

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$ x 3

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

Low degree siblings

16T408 x 15, 32T906 x 4, 32T907 x 4, 32T908 x 4, 32T1674 x 2, 32T1731 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, 2, 2, 2, 2, 1, 1, 1, 1 $ $8$ $2$ $( 3, 4)( 5,14)( 6,13)( 7,15)( 8,16)(11,12)$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $4$ $( 3, 7,12,16)( 4, 8,11,15)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $8$ $2$ $( 3, 8)( 4, 7)( 5,14)( 6,13)(11,16)(12,15)$
$ 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 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 2)( 3, 7)( 4, 8)( 5,13)( 6,14)( 9,10)(11,15)(12,16)$
$ 4, 4, 2, 2, 2, 2 $ $4$ $4$ $( 1, 2)( 3, 8,12,15)( 4, 7,11,16)( 5, 6)( 9,10)(13,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 2)( 3,11)( 4,12)( 5, 6)( 7,15)( 8,16)( 9,10)(13,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 3, 2, 4)( 5,15, 6,16)( 7,14, 8,13)( 9,11,10,12)$
$ 8, 8 $ $8$ $8$ $( 1, 3, 5,15,10,12,14, 8)( 2, 4, 6,16, 9,11,13, 7)$
$ 8, 8 $ $8$ $8$ $( 1, 3, 6, 7,10,12,13,16)( 2, 4, 5, 8, 9,11,14,15)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 3, 9,11)( 2, 4,10,12)( 5,15,13, 7)( 6,16,14, 8)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 3,10,12)( 2, 4, 9,11)( 5, 8,14,15)( 6, 7,13,16)$
$ 8, 8 $ $8$ $8$ $( 1, 3,13,16,10,12, 6, 7)( 2, 4,14,15, 9,11, 5, 8)$
$ 8, 8 $ $8$ $8$ $( 1, 3,14, 8,10,12, 5,15)( 2, 4,13, 7, 9,11, 6,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,13)(10,14)(11,15)(12,16)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 5,10,14)( 2, 6, 9,13)( 3, 8,12,15)( 4, 7,11,16)$
$ 4, 4, 2, 2, 2, 2 $ $4$ $4$ $( 1, 5,10,14)( 2, 6, 9,13)( 3,11)( 4,12)( 7,15)( 8,16)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 5,10,14)( 2, 6, 9,13)( 3,15,12, 8)( 4,16,11, 7)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 6,10,13)( 2, 5, 9,14)( 3, 7,12,16)( 4, 8,11,15)$
$ 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 $ $2$ $4$ $( 1, 6,10,13)( 2, 5, 9,14)( 3,16,12, 7)( 4,15,11, 8)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16)$
$ 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:  $128=2^{7}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [128, 734]
Character table: Data not available.