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Properties

Label 16T121
Order \(64\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $(C_2\times C_4).D_4$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $121$
Group :  $(C_2\times C_4).D_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,2)(3,4)(5,6)(7,14,12,10)(8,13,11,9)(15,16), (1,12,15,10,6,7,3,14)(2,11,16,9,5,8,4,13)
$|\Aut(F/K)|$:  $8$

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:  $C_4\wr C_2$ x 2, 32T41

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$, $D_{4}$ x 2

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

Low degree siblings

16T121 x 3, 32T129, 32T350

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

Group invariants

Order:  $64=2^{6}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [64, 20]
Character table: Data not available.