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Properties

Label 16T306
Order \(128\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $(C_2\times C_8).D_4$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $306$
Group :  $(C_2\times C_8).D_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $4$
Generators:  (1,5,4,8,2,6,3,7)(9,13,12,15,10,14,11,16), (1,11,6,15,4,10,7,13,2,12,5,16,3,9,8,14)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $D_{4}$ x 2, $C_8$ x 2, $C_4\times C_2$
16:  $C_8:C_2$, $C_2^2:C_4$, $C_8\times C_2$
32:  $(C_8:C_2):C_2$, $C_2^3 : C_4 $, $C_2^2 : C_8$
64:  16T84

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $C_8$

Low degree siblings

16T306, 32T706 x 2, 32T707, 32T708, 32T709 x 2, 32T710 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 $ $4$ $2$ $(13,14)(15,16)$
$ 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 $ $4$ $2$ $( 5, 6)( 7, 8)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $4$ $2$ $( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 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 $ $4$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)$
$ 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 $ $4$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,15,14,16)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,16,14,15)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,11,10,12)(13,16,14,15)$
$ 8, 8 $ $8$ $8$ $( 1, 5, 3, 7, 2, 6, 4, 8)( 9,13,11,16,10,14,12,15)$
$ 8, 8 $ $4$ $8$ $( 1, 5, 3, 7, 2, 6, 4, 8)( 9,13,12,15,10,14,11,16)$
$ 8, 8 $ $4$ $8$ $( 1, 5, 4, 8, 2, 6, 3, 7)( 9,13,11,16,10,14,12,15)$
$ 8, 8 $ $4$ $8$ $( 1, 7, 3, 6, 2, 8, 4, 5)( 9,15,12,14,10,16,11,13)$
$ 8, 8 $ $8$ $8$ $( 1, 7, 3, 6, 2, 8, 4, 5)( 9,15,11,13,10,16,12,14)$
$ 8, 8 $ $4$ $8$ $( 1, 7, 4, 5, 2, 8, 3, 6)( 9,15,11,13,10,16,12,14)$
$ 16 $ $8$ $16$ $( 1, 9, 5,13, 3,12, 7,15, 2,10, 6,14, 4,11, 8,16)$
$ 16 $ $8$ $16$ $( 1, 9, 5,13, 4,11, 8,16, 2,10, 6,14, 3,12, 7,15)$
$ 16 $ $8$ $16$ $( 1,11, 5,15, 3, 9, 7,14, 2,12, 6,16, 4,10, 8,13)$
$ 16 $ $8$ $16$ $( 1,11, 5,15, 4,10, 8,13, 2,12, 6,16, 3, 9, 7,14)$
$ 16 $ $8$ $16$ $( 1,13, 7, 9, 3,15, 6,12, 2,14, 8,10, 4,16, 5,11)$
$ 16 $ $8$ $16$ $( 1,13, 8,10, 4,16, 6,12, 2,14, 7, 9, 3,15, 5,11)$
$ 16 $ $8$ $16$ $( 1,15, 7,12, 3,14, 6,10, 2,16, 8,11, 4,13, 5, 9)$
$ 16 $ $8$ $16$ $( 1,15, 8,11, 4,13, 6,10, 2,16, 7,12, 3,14, 5, 9)$

Group invariants

Order:  $128=2^{7}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [128, 52]
Character table: Data not available.