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Properties

Label 16T79
Order \(64\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^5.C_2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $79$
Group :  $C_2^5.C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $2$
Generators:  (1,11,16,14)(2,12,15,13)(3,10,6,8)(4,9,5,7), (1,15)(2,16)(3,5)(4,6)(7,13)(8,14)(9,12)(10,11), (1,7,6,12)(2,8,5,11)(3,13,16,9)(4,14,15,10)
$|\Aut(F/K)|$:  $8$

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 12, $C_4\times C_2$ x 6, $C_2^3$
16:  $D_4\times C_2$ x 6, $C_2^2:C_4$ x 12, $C_4\times C_2^2$
32:  $C_2^2 \wr C_2$ x 4, $C_2 \times (C_2^2:C_4)$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$, $D_{4}$ x 6

Degree 8: $C_2^2:C_4$ x 3, $C_2^2 \wr C_2$ x 4

Low degree siblings

16T79 x 31, 32T72 x 12

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

Group invariants

Order:  $64=2^{6}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [64, 60]
Character table: Data not available.