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Properties

Label 16T205
Order \(128\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $(C_2\times C_4).C_2^4$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $205$
Group :  $(C_2\times C_4).C_2^4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (5,7,6,8)(9,11,10,12), (1,6,3,7,2,5,4,8)(9,15,12,13,10,16,11,14), (1,9)(2,10)(3,12)(4,11)(5,15)(6,16)(7,14)(8,13), (9,10)(11,12)(13,14)(15,16)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 15
4:  $C_4$ x 8, $C_2^2$ x 35
8:  $D_{4}$ x 4, $C_4\times C_2$ x 28, $C_2^3$ x 15
16:  $D_4\times C_2$ x 6, $Q_8:C_2$ x 2, $C_4\times C_2^2$ x 14, $C_2^4$
32:  $C_2 \times (C_4\times C_2):C_2$, $C_4 \times D_4$ x 4, $C_2^2 \times D_4$, 32T34
64:  32T204

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 8: $Q_8:C_2$

Low degree siblings

16T205, 32T442, 32T443 x 2, 32T444, 32T1959 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, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 5, 6)( 7, 8)(13,14)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 5, 6)( 7, 8)( 9,10)(11,12)$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $4$ $( 5, 7, 6, 8)( 9,11,10,12)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $4$ $4$ $( 5, 7, 6, 8)( 9,12,10,11)(13,14)(15,16)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $4$ $4$ $( 5, 8, 6, 7)( 9,11,10,12)(13,14)(15,16)$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $4$ $( 5, 8, 6, 7)( 9,12,10,11)$
$ 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, 2, 2, 2, 2 $ $2$ $4$ $( 1, 2)( 3, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,14)(15,16)$
$ 4, 4, 2, 2, 2, 2 $ $2$ $4$ $( 1, 2)( 3, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,14)(15,16)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,16,14,15)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,13)(10,14)(11,15)(12,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,14)(10,13)(11,16)(12,15)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,13,10,14)(11,15,12,16)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,14,10,13)(11,16,12,15)$
$ 8, 8 $ $4$ $8$ $( 1, 5, 3, 8, 2, 6, 4, 7)( 9,15,12,13,10,16,11,14)$
$ 8, 8 $ $4$ $8$ $( 1, 5, 3, 8, 2, 6, 4, 7)( 9,16,12,14,10,15,11,13)$
$ 8, 8 $ $4$ $8$ $( 1, 5, 4, 7, 2, 6, 3, 8)( 9,15,11,14,10,16,12,13)$
$ 8, 8 $ $4$ $8$ $( 1, 5, 4, 7, 2, 6, 3, 8)( 9,16,11,13,10,15,12,14)$
$ 8, 8 $ $4$ $8$ $( 1, 9, 3,12, 2,10, 4,11)( 5,13, 8,16, 6,14, 7,15)$
$ 8, 8 $ $4$ $8$ $( 1, 9, 4,11, 2,10, 3,12)( 5,13, 7,15, 6,14, 8,16)$
$ 8, 8 $ $4$ $8$ $( 1, 9, 3,12, 2,10, 4,11)( 5,14, 8,15, 6,13, 7,16)$
$ 8, 8 $ $4$ $8$ $( 1, 9, 4,11, 2,10, 3,12)( 5,14, 7,16, 6,13, 8,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,15)( 6,16)( 7,14)( 8,13)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,15, 6,16)( 7,14, 8,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,16)( 6,15)( 7,13)( 8,14)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,16, 6,15)( 7,13, 8,14)$
$ 8, 8 $ $2$ $8$ $( 1,13, 4,15, 2,14, 3,16)( 5, 9, 7,11, 6,10, 8,12)$
$ 8, 8 $ $2$ $8$ $( 1,13, 3,16, 2,14, 4,15)( 5, 9, 8,12, 6,10, 7,11)$
$ 8, 8 $ $2$ $8$ $( 1,13, 3,16, 2,14, 4,15)( 5,10, 8,11, 6, 9, 7,12)$
$ 8, 8 $ $2$ $8$ $( 1,13, 4,15, 2,14, 3,16)( 5,10, 7,12, 6, 9, 8,11)$
$ 8, 8 $ $4$ $8$ $( 1,13, 3,16, 2,14, 4,15)( 5,11, 7,10, 6,12, 8, 9)$
$ 8, 8 $ $4$ $8$ $( 1,13, 4,15, 2,14, 3,16)( 5,11, 8, 9, 6,12, 7,10)$
$ 8, 8 $ $4$ $8$ $( 1,13, 4,15, 2,14, 3,16)( 5,12, 8,10, 6,11, 7, 9)$
$ 8, 8 $ $4$ $8$ $( 1,13, 3,16, 2,14, 4,15)( 5,12, 7, 9, 6,11, 8,10)$
$ 8, 8 $ $2$ $8$ $( 1,15, 4,14, 2,16, 3,13)( 5,11, 7,10, 6,12, 8, 9)$
$ 8, 8 $ $2$ $8$ $( 1,15, 3,13, 2,16, 4,14)( 5,11, 8, 9, 6,12, 7,10)$
$ 8, 8 $ $2$ $8$ $( 1,15, 3,13, 2,16, 4,14)( 5,12, 8,10, 6,11, 7, 9)$
$ 8, 8 $ $2$ $8$ $( 1,15, 4,14, 2,16, 3,13)( 5,12, 7, 9, 6,11, 8,10)$

Group invariants

Order:  $128=2^{7}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [128, 1688]
Character table: Data not available.