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Properties

Label 16T92
Order \(64\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2\times C_2^2.D_4$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $92$
Group :  $C_2\times C_2^2.D_4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,14)(2,13)(3,8)(4,7)(5,10)(6,9)(11,15)(12,16), (1,3)(2,4)(5,8,14,16)(6,7,13,15)(9,11)(10,12), (1,2)(3,11)(4,12)(5,13)(6,14)(7,8)(9,10)(15,16)
$|\Aut(F/K)|$:  $4$

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 4, $C_4\times C_2$ x 6, $C_2^3$
16:  $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $C_4\times C_2^2$
32:  $C_2^3 : C_4 $ x 2, $C_2 \times (C_2^2:C_4)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7

Degree 8: $C_2^3$, $C_2^3: C_4$ x 2

Low degree siblings

16T76 x 4, 16T78, 16T93 x 2, 16T94 x 2, 16T102 x 2, 32T67 x 2, 32T70 x 2, 32T71, 32T94 x 2, 32T95

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

Group invariants

Order:  $64=2^{6}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [64, 90]
Character table: Data not available.