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Properties

Label 16T114
Order \(64\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $(C_4\times C_8):C_2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $114$
Group :  $(C_4\times C_8):C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,16)(2,15)(3,9)(4,10)(5,12)(6,11)(7,14)(8,13), (1,14)(2,13)(3,16)(4,15)(5,9)(6,10)(7,11)(8,12), (1,10,6,13,2,9,5,14)(3,11,7,15,4,12,8,16)
$|\Aut(F/K)|$:  $8$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_4$ x 4, $C_2^2$ x 7
8:  $D_{4}$ x 2, $C_4\times C_2$ x 6, $C_2^3$
16:  $D_4\times C_2$, $Q_8:C_2$, $C_4\times C_2^2$
32:  $C_4 \times D_4$

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

16T114, 32T119, 32T248 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 $ $2$ $2$ $( 9,10)(11,12)(13,14)(15,16)$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $4$ $( 9,13,10,14)(11,16,12,15)$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $4$ $( 9,14,10,13)(11,15,12,16)$
$ 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, 2, 2, 2, 2 $ $2$ $4$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,13,10,14)(11,16,12,15)$
$ 4, 4, 2, 2, 2, 2 $ $2$ $4$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,14,10,13)(11,15,12,16)$
$ 8, 8 $ $2$ $8$ $( 1, 3, 5, 8, 2, 4, 6, 7)( 9,11,13,16,10,12,14,15)$
$ 8, 8 $ $1$ $8$ $( 1, 3, 5, 8, 2, 4, 6, 7)( 9,12,13,15,10,11,14,16)$
$ 8, 8 $ $2$ $8$ $( 1, 3, 5, 8, 2, 4, 6, 7)( 9,15,14,12,10,16,13,11)$
$ 8, 8 $ $2$ $8$ $( 1, 3, 5, 8, 2, 4, 6, 7)( 9,16,14,11,10,15,13,12)$
$ 8, 8 $ $1$ $8$ $( 1, 4, 5, 7, 2, 3, 6, 8)( 9,11,13,16,10,12,14,15)$
$ 8, 8 $ $2$ $8$ $( 1, 4, 5, 7, 2, 3, 6, 8)( 9,15,14,12,10,16,13,11)$
$ 8, 8 $ $2$ $8$ $( 1, 4, 5, 7, 2, 3, 6, 8)( 9,16,14,11,10,15,13,12)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,13,10,14)(11,16,12,15)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,14,10,13)(11,15,12,16)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1, 6, 2, 5)( 3, 7, 4, 8)( 9,14,10,13)(11,15,12,16)$
$ 8, 8 $ $2$ $8$ $( 1, 7, 6, 4, 2, 8, 5, 3)( 9,15,14,12,10,16,13,11)$
$ 8, 8 $ $1$ $8$ $( 1, 7, 6, 4, 2, 8, 5, 3)( 9,16,14,11,10,15,13,12)$
$ 8, 8 $ $1$ $8$ $( 1, 8, 6, 3, 2, 7, 5, 4)( 9,15,14,12,10,16,13,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,13)( 6,14)( 7,16)( 8,15)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,13, 6,14)( 7,16, 8,15)$
$ 8, 8 $ $4$ $8$ $( 1, 9, 5,13, 2,10, 6,14)( 3,12, 8,15, 4,11, 7,16)$
$ 8, 8 $ $4$ $8$ $( 1, 9, 6,14, 2,10, 5,13)( 3,12, 7,16, 4,11, 8,15)$
$ 8, 8 $ $4$ $8$ $( 1,11, 6,15, 2,12, 5,16)( 3,14, 7,10, 4,13, 8, 9)$
$ 8, 8 $ $4$ $8$ $( 1,11, 5,16, 2,12, 6,15)( 3,14, 8, 9, 4,13, 7,10)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1,11, 2,12)( 3,14, 4,13)( 5,16, 6,15)( 7,10, 8, 9)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1,11)( 2,12)( 3,14)( 4,13)( 5,16)( 6,15)( 7,10)( 8, 9)$

Group invariants

Order:  $64=2^{6}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [64, 124]
Character table: Data not available.