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Properties

Label 16T608
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 $t$ :  $608$
Group :  $C_2^3.C_2^4.C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $4$
Generators:  (1,2)(3,4), (1,12,2,11)(3,10,4,9)(5,8)(6,7)(13,16)(14,15), (9,10)(11,12)(13,14)(15,16), (1,6)(2,5)(3,8)(4,7)(9,15)(10,16)(11,13)(12,14), (1,2)(5,6)(9,10)(13,14)
$|\Aut(F/K)|$:  $2$

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: $D_4$

Low degree siblings

16T477 x 4, 16T512 x 2, 16T519 x 2, 16T522 x 2, 16T608, 16T645 x 2, 16T654 x 2, 32T2376 x 2, 32T2377 x 4, 32T2378 x 4, 32T2379 x 4, 32T2380 x 2, 32T2542 x 2, 32T2543, 32T2544 x 2, 32T2545, 32T2546 x 2, 32T2547, 32T2548, 32T2583 x 2, 32T2584 x 2, 32T2585, 32T2586, 32T2587, 32T2588, 32T2595 x 2, 32T2596, 32T2597, 32T2598 x 2, 32T2599, 32T2600, 32T2978 x 2, 32T2979, 32T2980, 32T2981, 32T2982 x 2, 32T2983, 32T2984 x 2, 32T2985 x 2, 32T3153, 32T3154 x 2, 32T3155, 32T3156, 32T3157, 32T3197, 32T3198, 32T3199, 32T3200, 32T4765 x 2, 32T7444, 32T7471

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

Group invariants

Order:  $256=2^{8}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [256, 26539]
Character table: Data not available.