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Properties

Label 16T123
Order \(64\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $(C_2\times OD_{16}).C_2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $123$
Group :  $(C_2\times OD_{16}).C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,7)(2,8)(3,6)(4,5)(9,16,10,15)(11,14,12,13), (1,11)(2,12)(3,9)(4,10)(5,13,6,14)(7,15,8,16)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 6, $C_2^2$
8:  $D_{4}$ x 3, $C_4\times C_2$ x 3, $Q_8$
16:  $C_2^2:C_4$ x 3, $C_4^2$, $C_4:C_4$ x 3
32:  32T41

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

Degree 8: $D_4$

Low degree siblings

16T74 x 4, 32T64 x 2, 32T131

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

Group invariants

Order:  $64=2^{6}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [64, 18]
Character table: Data not available.