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Properties

Label 16T657
Order \(256\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^3.C_2^4.C_2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $657$
Group :  $C_2^3.C_2^4.C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $4$
Generators:  (1,2)(3,4)(9,11,10,12)(13,15,14,16), (1,3)(2,4)(5,8)(6,7)(9,13,10,14)(11,15,12,16), (9,10)(11,12)(13,14)(15,16), (1,6)(2,5)(3,8)(4,7)(11,12)(13,14), (1,10,2,9)(3,14)(4,13)(5,11)(6,12)(7,15,8,16)
$|\Aut(F/K)|$:  $2$

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, $C_2^4$
32:  $C_2^2 \wr C_2$ x 4, $C_2^2 \times D_4$ x 3
64:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 2, 16T105
128:  16T245

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 8: $D_4\times C_2$

Low degree siblings

16T636 x 4, 16T652 x 4, 16T657 x 3, 32T3110 x 4, 32T3111 x 2, 32T3112 x 2, 32T3113 x 4, 32T3114 x 2, 32T3115 x 2, 32T3116 x 2, 32T3185 x 2, 32T3186 x 2, 32T3187 x 4, 32T3188 x 2, 32T3189 x 2, 32T3190 x 2, 32T3209 x 2, 32T3210 x 2, 32T3211 x 2, 32T3212 x 2, 32T3213 x 2, 32T4769 x 4, 32T7442 x 2, 32T7470 x 2

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

Group invariants

Order:  $256=2^{8}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [256, 26545]
Character table: Data not available.