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Properties

Label 16T610
Order \(256\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^2.C_2^5.C_2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $610$
Group :  $C_2^2.C_2^5.C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $4$
Generators:  (1,8,12,16,2,7,11,15)(3,5,9,13,4,6,10,14), (1,7,4,5,2,8,3,6)(9,13,11,15,10,14,12,16), (1,12,2,11)(3,9,4,10)(5,8,6,7)(13,16,14,15)
$|\Aut(F/K)|$:  $4$

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

16T503 x 4, 16T610 x 7, 32T2505 x 4, 32T2506 x 4, 32T2507 x 2, 32T2508 x 4, 32T2509 x 2, 32T2990 x 4, 32T2991 x 4, 32T2992 x 4, 32T2993 x 4, 32T2994 x 4, 32T2995 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 $ $4$ $2$ $(13,14)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 5, 6)( 7, 8)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $4$ $2$ $( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $4$ $( 5,15, 6,16)( 7,14, 8,13)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 5,15)( 6,16)( 7,14)( 8,13)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $8$ $4$ $( 5,15, 6,16)( 7,14, 8,13)( 9,10)(11,12)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $8$ $2$ $( 5,15)( 6,16)( 7,14)( 8,13)( 9,10)(11,12)$
$ 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 $ $4$ $4$ $( 1, 2)( 3, 4)( 5,15, 6,16)( 7,14, 8,13)( 9,10)(11,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 2)( 3, 4)( 5,15)( 6,16)( 7,14)( 8,13)( 9,10)(11,12)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)$
$ 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 $ $4$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,16,14,15)$
$ 4, 4, 4, 4 $ $2$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,16,14,15)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,12,10,11)(13,16,14,15)$
$ 4, 4, 2, 2, 2, 2 $ $4$ $4$ $( 1, 3, 2, 4)( 5,13)( 6,14)( 7,15)( 8,16)( 9,11,10,12)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 3, 2, 4)( 5,13, 6,14)( 7,15, 8,16)( 9,11,10,12)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 3, 2, 4)( 5,13)( 6,14)( 7,15)( 8,16)( 9,12,10,11)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 3, 2, 4)( 5,13, 6,14)( 7,15, 8,16)( 9,12,10,11)$
$ 4, 4, 4, 4 $ $1$ $4$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,12,10,11)(13,16,14,15)$
$ 4, 4, 2, 2, 2, 2 $ $4$ $4$ $( 1, 4, 2, 3)( 5,13)( 6,14)( 7,15)( 8,16)( 9,12,10,11)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 4, 2, 3)( 5,13, 6,14)( 7,15, 8,16)( 9,12,10,11)$
$ 8, 8 $ $8$ $8$ $( 1, 5, 3, 7, 2, 6, 4, 8)( 9,15,11,14,10,16,12,13)$
$ 8, 8 $ $16$ $8$ $( 1, 5, 3, 7, 2, 6, 4, 8)( 9,15,12,13,10,16,11,14)$
$ 8, 8 $ $8$ $8$ $( 1, 5, 4, 8, 2, 6, 3, 7)( 9,15,12,13,10,16,11,14)$
$ 8, 8 $ $16$ $8$ $( 1, 5,11,14, 2, 6,12,13)( 3, 7,10,16, 4, 8, 9,15)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 5,11,14)( 2, 6,12,13)( 3, 7,10,16)( 4, 8, 9,15)$
$ 8, 8 $ $8$ $8$ $( 1, 7, 3, 6, 2, 8, 4, 5)( 9,13,11,15,10,14,12,16)$
$ 8, 8 $ $16$ $8$ $( 1, 7, 3, 6, 2, 8, 4, 5)( 9,13,12,16,10,14,11,15)$
$ 8, 8 $ $8$ $8$ $( 1, 7, 4, 5, 2, 8, 3, 6)( 9,13,12,16,10,14,11,15)$
$ 8, 8 $ $16$ $8$ $( 1, 7,11,15, 2, 8,12,16)( 3, 6,10,14, 4, 5, 9,13)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 7,11,15)( 2, 8,12,16)( 3, 6,10,14)( 4, 5, 9,13)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,15, 6,16)( 7,14, 8,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,15)( 6,16)( 7,14)( 8,13)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 9, 2,10)( 3,11, 4,12)( 5,15, 6,16)( 7,14, 8,13)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1,11, 2,12)( 3,10, 4, 9)( 5,13)( 6,14)( 7,15)( 8,16)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1,11, 2,12)( 3,10, 4, 9)( 5,13, 6,14)( 7,15, 8,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1,11)( 2,12)( 3,10)( 4, 9)( 5,13)( 6,14)( 7,15)( 8,16)$

Group invariants

Order:  $256=2^{8}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [256, 5753]
Character table: Data not available.