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Properties

Label 16T68
Order \(64\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^3:(C_2\times C_4)$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $68$
Group :  $C_2^3:(C_2\times C_4)$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,16)(2,15)(3,5)(4,6), (1,10,2,9)(3,11,4,12)(5,14,6,13)(7,15,8,16), (1,6,2,5)(3,16,4,15)(7,12,8,11)(9,14,10,13), (7,10)(8,9)(11,14)(12,13), (1,16)(2,15)(7,10)(8,9)
$|\Aut(F/K)|$:  $4$

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:  $C_4\times C_2$ x 28, $C_2^3$ x 15
16:  $C_4\times C_2^2$ x 14, $C_2^4$
32:  $C_2^3 : D_4 $ x 2, 32T34

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

Degree 8: $C_4\times C_2$, $C_2^3 : D_4 $ x 2

Low degree siblings

16T68 x 7, 32T53 x 8, 32T54 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 $ $2$ $2$ $( 7,10)( 8, 9)(11,14)(12,13)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 3, 5)( 4, 6)(11,14)(12,13)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 3, 5)( 4, 6)( 7,10)( 8, 9)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 9)( 8,10)(11,13)(12,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 2)( 3, 6)( 4, 5)( 7, 8)( 9,10)(11,13)(12,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 2)( 3, 6)( 4, 5)( 7, 9)( 8,10)(11,12)(13,14)(15,16)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 3, 2, 4)( 5,15, 6,16)( 7,11, 8,12)( 9,13,10,14)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 3, 2, 4)( 5,15, 6,16)( 7,14, 8,13)( 9,12,10,11)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 3,15, 6)( 2, 4,16, 5)( 7,11, 9,13)( 8,12,10,14)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 3,15, 6)( 2, 4,16, 5)( 7,14, 9,12)( 8,13,10,11)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 4, 2, 3)( 5,16, 6,15)( 7,12, 8,11)( 9,14,10,13)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 4, 2, 3)( 5,16, 6,15)( 7,13, 8,14)( 9,11,10,12)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 4,15, 5)( 2, 3,16, 6)( 7,12, 9,14)( 8,11,10,13)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 4,15, 5)( 2, 3,16, 6)( 7,13, 9,11)( 8,14,10,12)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 7, 2, 8)( 3,11, 4,12)( 5,14, 6,13)( 9,16,10,15)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 7,15, 9)( 2, 8,16,10)( 3,11, 6,13)( 4,12, 5,14)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 7, 2, 8)( 3,14, 4,13)( 5,11, 6,12)( 9,16,10,15)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 7,15, 9)( 2, 8,16,10)( 3,14, 6,12)( 4,13, 5,11)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 8, 2, 7)( 3,12, 4,11)( 5,13, 6,14)( 9,15,10,16)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 8,15,10)( 2, 7,16, 9)( 3,12, 6,14)( 4,11, 5,13)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 8, 2, 7)( 3,13, 4,14)( 5,12, 6,11)( 9,15,10,16)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 8,15,10)( 2, 7,16, 9)( 3,13, 6,11)( 4,14, 5,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1,11)( 2,12)( 3, 8)( 4, 7)( 5, 9)( 6,10)(13,15)(14,16)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1,11,16,14)( 2,12,15,13)( 3, 8, 5, 9)( 4, 7, 6,10)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1,11,16,14)( 2,12,15,13)( 3, 9, 5, 8)( 4,10, 6, 7)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1,11)( 2,12)( 3, 9)( 4,10)( 5, 8)( 6, 7)(13,15)(14,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1,12)( 2,11)( 3, 7)( 4, 8)( 5,10)( 6, 9)(13,16)(14,15)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1,12,16,13)( 2,11,15,14)( 3, 7, 5,10)( 4, 8, 6, 9)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1,12,16,13)( 2,11,15,14)( 3,10, 5, 7)( 4, 9, 6, 8)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1,12)( 2,11)( 3,10)( 4, 9)( 5, 7)( 6, 8)(13,16)(14,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,15)( 2,16)( 3, 6)( 4, 5)( 7, 9)( 8,10)(11,13)(12,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,16)( 2,15)( 3, 5)( 4, 6)( 7,10)( 8, 9)(11,14)(12,13)$

Group invariants

Order:  $64=2^{6}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [64, 199]
Character table: Data not available.