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Properties

Label 16T1223
Order \(1024\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $1223$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $7$
Generators:  (1,8,14,4,10,15,6,11,2,7,13,3,9,16,5,12), (1,15,14,11,9,8,5,3,2,16,13,12,10,7,6,4)
$|\Aut(F/K)|$:  $2$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $D_{4}$ x 2, $C_8$ x 2, $C_4\times C_2$
16:  $C_8:C_2$, $C_2^2:C_4$, $C_8\times C_2$
32:  $(C_8:C_2):C_2$, $C_2^3 : C_4 $, $C_2^2 : C_8$
64:  $((C_8 : C_2):C_2):C_2$ x 2, 16T84
128:  16T228
256:  16T565
512:  16T817

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $C_8$

Low degree siblings

16T1155 x 8, 16T1223 x 7, 32T35849 x 8, 32T35850 x 16, 32T35851 x 8, 32T35852 x 8, 32T35853 x 16, 32T35854 x 4, 32T35855 x 4, 32T36405 x 32, 32T36406 x 8, 32T36407 x 32, 32T36408 x 16, 32T36409 x 8, 32T36410 x 16, 32T36411 x 4, 32T36412 x 16, 32T36413 x 16, 32T36414 x 16, 32T36415 x 16, 32T36416 x 16, 32T36417 x 16, 32T36418 x 8, 32T36419 x 4, 32T36420 x 8, 32T36421 x 16, 32T48622 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, 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, 4, 4 $ $8$ $4$ $( 1,10, 2, 9)( 3,12, 4,11)( 5,14, 6,13)( 7,16, 8,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 5, 6)( 7, 8)(13,14)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 3, 4)( 7, 8)(11,12)(15,16)$
$ 8, 8 $ $32$ $8$ $( 1,14,10, 6, 2,13, 9, 5)( 3,16,12, 8, 4,15,11, 7)$
$ 8, 8 $ $32$ $8$ $( 1, 6, 9,14, 2, 5,10,13)( 3, 8,11,16, 4, 7,12,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $16$ $2$ $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 3, 4)( 7, 8)( 9,10)(13,14)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 7, 8)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $4$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 9,10)(11,12)(13,14)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1,10, 2, 9)( 3,12, 4,11)( 5,14, 6,13)( 7,15, 8,16)$
$ 16 $ $64$ $16$ $( 1, 8,14, 4,10,15, 6,11, 2, 7,13, 3, 9,16, 5,12)$
$ 16 $ $64$ $16$ $( 1,15,13,12,10, 7, 5, 4, 2,16,14,11, 9, 8, 6, 3)$
$ 16 $ $64$ $16$ $( 1, 4, 6, 7, 9,12,14,15, 2, 3, 5, 8,10,11,13,16)$
$ 16 $ $64$ $16$ $( 1,11, 5,15, 9, 4,13, 8, 2,12, 6,16,10, 3,14, 7)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $(11,12)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $8$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(13,14)$
$ 4, 4, 2, 2, 2, 2 $ $32$ $4$ $( 1,10, 2, 9)( 3,12)( 4,11)( 5,14, 6,13)( 7,16)( 8,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 5, 6)( 7, 8)(11,12)(13,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 1, 2)( 3, 4)( 9,10)(15,16)$
$ 8, 8 $ $32$ $8$ $( 1,14,10, 6, 2,13, 9, 5)( 3,16,11, 8, 4,15,12, 7)$
$ 8, 8 $ $32$ $8$ $( 1, 6, 9,14, 2, 5,10,13)( 3, 8,11,15, 4, 7,12,16)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $8$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$
$ 4, 4, 2, 2, 2, 2 $ $32$ $4$ $( 1,10, 2, 9)( 3,12, 4,11)( 5,14)( 6,13)( 7,16)( 8,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 3, 4)( 7, 8)(11,12)(13,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 1, 2)( 5, 6)( 9,10)(15,16)$
$ 4, 4, 4, 4 $ $64$ $4$ $( 1,14, 9, 5)( 2,13,10, 6)( 3,16,11, 7)( 4,15,12, 8)$
$ 4, 4, 4, 4 $ $64$ $4$ $( 1, 6, 9,14)( 2, 5,10,13)( 3, 8,11,16)( 4, 7,12,15)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 7, 8)(13,14)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $8$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 9,10)(11,12)(15,16)$
$ 4, 4, 2, 2, 2, 2 $ $32$ $4$ $( 1,10, 2, 9)( 3,12, 4,11)( 5,14)( 6,13)( 7,15)( 8,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 3, 4)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 1, 2)( 5, 6)( 7, 8)( 9,10)$
$ 16 $ $64$ $16$ $( 1, 8,14, 3, 9,16, 6,11, 2, 7,13, 4,10,15, 5,12)$
$ 16 $ $64$ $16$ $( 1,15,14,12,10, 7, 5, 4, 2,16,13,11, 9, 8, 6, 3)$
$ 16 $ $64$ $16$ $( 1, 4, 6, 7, 9,12,14,16, 2, 3, 5, 8,10,11,13,15)$
$ 16 $ $64$ $16$ $( 1,11, 5,15,10, 3,14, 8, 2,12, 6,16, 9, 4,13, 7)$

Group invariants

Order:  $1024=2^{10}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  Data not available
Character table: Data not available.