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Properties

Label 16T464
Order \(256\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^4.C_2^4$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $464$
Group :  $C_2^4.C_2^4$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,2)(3,4), (1,6,4,7)(2,5,3,8)(9,16,12,13)(10,15,11,14), (1,12)(2,11)(3,10)(4,9)(5,13)(6,14)(7,15)(8,16), (9,10)(11,12)(13,14)(15,16), (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:  $C_4$ x 8, $C_2^2$ x 35
8:  $D_{4}$ x 8, $C_4\times C_2$ x 28, $C_2^3$ x 15
16:  $D_4\times C_2$ x 12, $C_2^2:C_4$ x 16, $C_4\times C_2^2$ x 14, $C_2^4$
32:  $C_2^3 : D_4 $ x 4, $C_2 \times (C_2^2:C_4)$ x 12, $C_2^2 \times D_4$ x 2, 32T34
64:  16T68 x 2, 16T87 x 4, 32T262
128:  32T992

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$

Degree 8: $C_4\times C_2$

Low degree siblings

16T454 x 8, 16T458 x 4, 16T459 x 4, 16T463 x 2, 16T464, 32T2256 x 8, 32T2257 x 8, 32T2273 x 4, 32T2274 x 4, 32T2275 x 8, 32T2276 x 2, 32T2277 x 2, 32T2278 x 4, 32T2279 x 2, 32T2280 x 4, 32T2281 x 2, 32T2298 x 4, 32T2299, 32T2300, 32T2301 x 2, 32T2302 x 2, 32T2303, 32T2304 x 2, 32T2305 x 4, 32T2306 x 4, 32T2307, 32T7268 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 $ $2$ $2$ $( 9,10)(11,12)(13,14)(15,16)$
$ 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, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 5, 6)( 7, 8)( 9,10)(11,12)$
$ 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 $ $4$ $2$ $( 3, 4)( 7, 8)(11,12)(13,14)$
$ 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, 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 $ $2$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,15)(14,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,16)(14,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,15)(14,16)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,15,14,16)$
$ 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 $ $4$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 5, 3, 7)( 2, 6, 4, 8)( 9,15,11,13)(10,16,12,14)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 5, 3, 7)( 2, 6, 4, 8)( 9,15,12,14)(10,16,11,13)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 5, 3, 8)( 2, 6, 4, 7)( 9,15,12,13)(10,16,11,14)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 5, 3, 8)( 2, 6, 4, 7)( 9,15,11,14)(10,16,12,13)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 7, 3, 5)( 2, 8, 4, 6)( 9,13,11,15)(10,14,12,16)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 7, 3, 5)( 2, 8, 4, 6)( 9,13,12,16)(10,14,11,15)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 7, 4, 5)( 2, 8, 3, 6)( 9,13,11,16)(10,14,12,15)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 7, 4, 5)( 2, 8, 3, 6)( 9,13,12,15)(10,14,11,16)$
$ 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 $ $4$ $4$ $( 1, 9, 2,10)( 3,12, 4,11)( 5,15, 6,16)( 7,14, 8,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1,11)( 2,12)( 3, 9)( 4,10)( 5,13)( 6,14)( 7,15)( 8,16)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1,11)( 2,12)( 3, 9)( 4,10)( 5,13, 6,14)( 7,15, 8,16)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1,11, 2,12)( 3, 9, 4,10)( 5,13, 6,14)( 7,15, 8,16)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1,11, 2,12)( 3,10, 4, 9)( 5,13)( 6,14)( 7,16)( 8,15)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1,11, 2,12)( 3,10, 4, 9)( 5,13, 6,14)( 7,16, 8,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1,11)( 2,12)( 3,10)( 4, 9)( 5,13)( 6,14)( 7,16)( 8,15)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1,13, 3,15)( 2,14, 4,16)( 5, 9, 7,11)( 6,10, 8,12)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1,13, 4,16)( 2,14, 3,15)( 5, 9, 7,11)( 6,10, 8,12)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1,13, 3,16)( 2,14, 4,15)( 5, 9, 7,12)( 6,10, 8,11)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1,13, 4,15)( 2,14, 3,16)( 5, 9, 7,12)( 6,10, 8,11)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1,15, 3,13)( 2,16, 4,14)( 5,11, 7, 9)( 6,12, 8,10)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1,15, 4,14)( 2,16, 3,13)( 5,11, 7, 9)( 6,12, 8,10)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1,15, 4,13)( 2,16, 3,14)( 5,11, 8, 9)( 6,12, 7,10)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1,15, 3,14)( 2,16, 4,13)( 5,11, 8, 9)( 6,12, 7,10)$

Group invariants

Order:  $256=2^{8}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [256, 14764]
Character table: Data not available.