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Properties

Label 16T528
Order \(256\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^6.C_2^2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $528$
Group :  $C_2^6.C_2^2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,5)(2,6), (1,7,2,8)(3,6,4,5)(9,12,10,11)(13,15,14,16), (1,2)(3,4)(5,6)(7,8), (1,9,2,10)(3,15,4,16)(5,14,6,13)(7,11,8,12)
$|\Aut(F/K)|$:  $4$

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 12, $C_2^3$ x 15
16:  $D_4\times C_2$ x 18, $Q_8:C_2$ x 4, $C_2^4$
32:  $C_2^2 \wr C_2$ x 4, $C_2^3 : D_4 $ x 2, $C_2 \times (C_4\times C_2):C_2$ x 2, $C_2^2 \times D_4$ x 3
64:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 4, 16T73, 16T105, 16T115 x 4, 16T117
128:  16T245 x 2, 32T1074

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 8: $Q_8:C_2$, $(((C_4 \times C_2): C_2):C_2):C_2$ x 2

Low degree siblings

16T473 x 32, 16T528 x 31, 32T2353 x 64, 32T2354 x 64, 32T2355 x 16, 32T2356 x 16, 32T2357 x 8, 32T2358 x 16, 32T2621 x 8, 32T2622 x 8, 32T2623 x 8, 32T2624 x 16, 32T2625 x 16, 32T2626 x 8, 32T4755 x 16, 32T7067 x 4

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

Group invariants

Order:  $256=2^{8}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [256, 16972]
Character table: Data not available.