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Properties

Label 16T646
Order \(256\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^4.C_2^3.C_2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $646$
Group :  $C_2^4.C_2^3.C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $4$
Generators:  (1,9,8,15)(2,10,7,16)(3,13,5,12)(4,14,6,11), (1,3,2,4)(5,8,6,7)(9,16)(10,15)(11,14)(12,13), (1,11,4,15,2,12,3,16)(5,10,7,13,6,9,8,14)
$|\Aut(F/K)|$:  $2$

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 12, $C_4\times C_2$ x 6, $C_2^3$
16:  $D_4\times C_2$ x 6, $C_2^2:C_4$ x 12, $C_4\times C_2^2$
32:  $C_2^2 \wr C_2$ x 4, $C_2 \times (C_2^2:C_4)$ x 3
64:  $(((C_4 \times C_2): C_2):C_2):C_2$, 16T79, 16T146
128:  32T1151

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$, $D_{4}$ x 2

Degree 8: $C_2^2:C_4$

Low degree siblings

16T549 x 4, 16T631 x 4, 16T646 x 3, 32T2722 x 2, 32T2723 x 2, 32T2724 x 4, 32T2725 x 2, 32T2726 x 4, 32T2727 x 2, 32T2728 x 2, 32T3083 x 2, 32T3084 x 2, 32T3085 x 2, 32T3086 x 2, 32T3087 x 2, 32T3088 x 4, 32T3158 x 2, 32T3159 x 2, 32T3160 x 2, 32T3161 x 2, 32T3162 x 2, 32T7431 x 2, 32T7449 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 $ $4$ $2$ $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,15)(10,16)(11,14)(12,13)$
$ 8, 8 $ $16$ $8$ $( 1, 9, 3,13, 2,10, 4,14)( 5,11, 8,16, 6,12, 7,15)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 9, 7,15)( 2,10, 8,16)( 3,13, 5,11)( 4,14, 6,12)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 9, 8,16)( 2,10, 7,15)( 3,13, 6,12)( 4,14, 5,11)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 9, 3,13)( 2,10, 4,14)( 5,12, 7,16)( 6,11, 8,15)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 9, 7,16)( 2,10, 8,15)( 3,13, 6,11)( 4,14, 5,12)$
$ 8, 8 $ $16$ $8$ $( 1,13, 4,10, 2,14, 3, 9)( 5,15, 7,11, 6,16, 8,12)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1,13, 7,11)( 2,14, 8,12)( 3, 9, 5,15)( 4,10, 6,16)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1,13, 8,12)( 2,14, 7,11)( 3, 9, 6,16)( 4,10, 5,15)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1,13, 3, 9)( 2,14, 4,10)( 5,16, 7,12)( 6,15, 8,11)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1,13, 8,11)( 2,14, 7,12)( 3, 9, 5,16)( 4,10, 6,15)$

Group invariants

Order:  $256=2^{8}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [256, 5755]
Character table: Data not available.