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Properties

Label 16T117
Order \(64\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_4^2:C_2^2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $117$
Group :  $C_4^2:C_2^2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (9,12)(10,11)(13,16)(14,15), (1,15)(2,16)(3,10)(4,9)(5,11)(6,12)(7,13)(8,14), (1,2)(3,4)(5,6)(7,8), (1,6,2,5)(3,8,4,7)(9,16,10,15)(11,14,12,13)
$|\Aut(F/K)|$:  $8$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 15
4:  $C_2^2$ x 35
8:  $D_{4}$ x 4, $C_2^3$ x 15
16:  $D_4\times C_2$ x 6, $Q_8:C_2$ x 4, $C_2^4$
32:  $C_2 \times (C_4\times C_2):C_2$ x 2, $C_2^2 \times D_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 8: $D_4\times C_2$, $Q_8:C_2$ x 2

Low degree siblings

16T117 x 7, 32T125 x 4, 32T126 x 2, 32T236 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 $ $2$ $2$ $( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 9,11)(10,12)(13,15)(14,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 9,12)(10,11)(13,16)(14,15)$
$ 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 $ $2$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,11)(10,12)(13,15)(14,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12)(10,11)(13,16)(14,15)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,13,10,14)(11,15,12,16)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,14,10,13)(11,16,12,15)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,15,10,16)(11,13,12,14)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,16,10,15)(11,14,12,13)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,13,10,14)(11,15,12,16)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,14,10,13)(11,16,12,15)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,16,10,15)(11,14,12,13)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,13,10,14)(11,15,12,16)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1, 5, 2, 6)( 3, 7, 4, 8)( 9,14,10,13)(11,16,12,15)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1, 6, 2, 5)( 3, 8, 4, 7)( 9,13,10,14)(11,15,12,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,11)(10,12)(13,15)(14,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,12)(10,11)(13,16)(14,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 8)( 2, 7)( 3, 5)( 4, 6)( 9,12)(10,11)(13,16)(14,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 9)( 2,10)( 3,15)( 4,16)( 5,14)( 6,13)( 7,11)( 8,12)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 9, 2,10)( 3,15, 4,16)( 5,14, 6,13)( 7,11, 8,12)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 9, 7,11)( 2,10, 8,12)( 3,15, 6,13)( 4,16, 5,14)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 9, 8,12)( 2,10, 7,11)( 3,15, 5,14)( 4,16, 6,13)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1,13, 8,16)( 2,14, 7,15)( 3,12, 5, 9)( 4,11, 6,10)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1,13, 7,15)( 2,14, 8,16)( 3,12, 6,10)( 4,11, 5, 9)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1,13)( 2,14)( 3,12)( 4,11)( 5, 9)( 6,10)( 7,15)( 8,16)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1,13, 2,14)( 3,12, 4,11)( 5, 9, 6,10)( 7,15, 8,16)$

Group invariants

Order:  $64=2^{6}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [64, 206]
Character table: Data not available.