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Properties

Label 16T520
Order \(256\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_4:Q_8.C_2^3$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $520$
Group :  $C_4:Q_8.C_2^3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $5$
Generators:  (1,7,11,14,2,8,12,13)(3,5,9,16,4,6,10,15), (1,16,9,5,2,15,10,6)(3,14,12,7,4,13,11,8), (1,9)(2,10)(3,12)(4,11)(5,15)(6,16)(7,13)(8,14)
$|\Aut(F/K)|$:  $2$

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, $C_2 \times (C_2^2:C_4)$
64:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 2, 16T76
128:  16T235

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $C_2^3: C_4$

Low degree siblings

16T520 x 3, 32T2589 x 2, 32T2590 x 4, 32T2591 x 4, 32T2592 x 4, 32T6715 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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

Group invariants

Order:  $256=2^{8}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [256, 6579]
Character table: Data not available.