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Properties

Label 16T1194
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 $t$ :  $1194$
Parity:  $1$
Primitive:  No
Nilpotency class:  $7$
Generators:  (1,12,5,16,9,3,14,7)(2,11,6,15,10,4,13,8), (1,14,10,5,2,13,9,6)(3,15,11,7,4,16,12,8)
$|\Aut(F/K)|$:  $2$

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

16T1194 x 7, 16T1216 x 8, 32T36186 x 16, 32T36187 x 16, 32T36188 x 16, 32T36189 x 16, 32T36190 x 16, 32T36191 x 8, 32T36192 x 32, 32T36193 x 16, 32T36194 x 16, 32T36195 x 32, 32T36196 x 8, 32T36197 x 16, 32T36198 x 16, 32T36199 x 4, 32T36200 x 8, 32T36201 x 8, 32T36202 x 4, 32T36203 x 8, 32T36350 x 16, 32T36351 x 8, 32T36352 x 8, 32T36353 x 16, 32T36354 x 4, 32T36355 x 4, 32T50842 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 $ $8$ $2$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 3, 4)( 5, 6)(11,12)(13,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 3, 4)( 7, 8)(11,12)(15,16)$
$ 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, 4, 4 $ $32$ $4$ $( 1, 5, 9,14)( 2, 6,10,13)( 3, 7,12,16)( 4, 8,11,15)$
$ 4, 4, 4, 4 $ $32$ $4$ $( 1,14, 9, 5)( 2,13,10, 6)( 3,16,12, 7)( 4,15,11, 8)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 7, 8)( 9,10)(11,12)(13,14)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,14, 6,13)( 7,15, 8,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 5, 6)( 9,10)(11,12)(15,16)$
$ 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, 2, 2 $ $8$ $2$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,14)( 6,13)( 7,15)( 8,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $4$ $2$ $( 3, 4)( 5, 6)( 7, 8)(11,12)(13,14)(15,16)$
$ 8, 8 $ $64$ $8$ $( 1,12, 5,16, 9, 3,14, 7)( 2,11, 6,15,10, 4,13, 8)$
$ 8, 8 $ $64$ $8$ $( 1, 3, 5, 7, 9,12,14,16)( 2, 4, 6, 8,10,11,13,15)$
$ 8, 8 $ $64$ $8$ $( 1,16,14,12, 9, 7, 5, 3)( 2,15,13,11,10, 8, 6, 4)$
$ 8, 8 $ $64$ $8$ $( 1, 7,14, 3, 9,16, 5,12)( 2, 8,13, 4,10,15, 6,11)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $(11,12)(15,16)$
$ 4, 4, 2, 2, 2, 2 $ $32$ $4$ $( 1, 9)( 2,10)( 3,12, 4,11)( 5,14)( 6,13)( 7,16, 8,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 3, 4)( 5, 6)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $8$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(13,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 1, 2)( 7, 8)( 9,10)(11,12)$
$ 4, 4, 4, 4 $ $32$ $4$ $( 1, 5, 9,14)( 2, 6,10,13)( 3, 7,12,15)( 4, 8,11,16)$
$ 4, 4, 4, 4 $ $32$ $4$ $( 1,14, 9, 5)( 2,13,10, 6)( 3,16,11, 7)( 4,15,12, 8)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $(13,14)(15,16)$
$ 4, 4, 2, 2, 2, 2 $ $32$ $4$ $( 1, 9)( 2,10)( 3,12)( 4,11)( 5,14, 6,13)( 7,16, 8,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 3, 4)( 5, 6)(11,12)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $8$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 1, 2)( 7, 8)( 9,10)(13,14)$
$ 8, 8 $ $64$ $8$ $( 1, 5, 9,14, 2, 6,10,13)( 3, 7,12,16, 4, 8,11,15)$
$ 8, 8 $ $64$ $8$ $( 1,14,10, 6, 2,13, 9, 5)( 3,16,11, 8, 4,15,12, 7)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 7, 8)( 9,10)(11,12)(15,16)$
$ 4, 4, 2, 2, 2, 2 $ $32$ $4$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,14)( 6,13)( 7,15)( 8,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $8$ $2$ $( 3, 4)( 5, 6)( 7, 8)( 9,10)(13,14)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 5, 6)( 9,10)(11,12)(13,14)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 3, 4)( 9,10)$
$ 8, 8 $ $64$ $8$ $( 1,12, 5,16,10, 4,13, 7)( 2,11, 6,15, 9, 3,14, 8)$
$ 8, 8 $ $64$ $8$ $( 1, 3, 5, 7, 9,12,14,15)( 2, 4, 6, 8,10,11,13,16)$
$ 8, 8 $ $64$ $8$ $( 1,16,13,12, 9, 7, 5, 3)( 2,15,14,11,10, 8, 6, 4)$
$ 8, 8 $ $64$ $8$ $( 1, 7,14, 4,10,15, 5,12)( 2, 8,13, 3, 9,16, 6,11)$

Group invariants

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