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Properties

Label 16T456
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 $t$ :  $456$
Group :  $C_2^3.C_2^4.C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $4$
Generators:  (5,6)(7,8)(13,14)(15,16), (9,10)(11,12)(13,14)(15,16), (1,9)(2,10)(3,11,4,12)(5,13)(6,14)(7,16,8,15), (1,3,2,4)(5,8,6,7)(13,14)(15,16), (1,5)(2,6)(3,8)(4,7)(9,14)(10,13)(11,16)(12,15)
$|\Aut(F/K)|$:  $2$

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 : C_4 $ x 4, $C_2 \times (C_2^2:C_4)$ x 12, $C_2^2 \times D_4$ x 2, 32T34
64:  16T76 x 6, 32T262
128:  32T1107

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

16T456 x 7, 16T467 x 4, 32T2262 x 4, 32T2263 x 4, 32T2264 x 8, 32T2265 x 4, 32T2266 x 4, 32T2267 x 4, 32T2268 x 4, 32T2317 x 2, 32T2318 x 2, 32T2319 x 4, 32T2320 x 2, 32T7416 x 8

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

Group invariants

Order:  $256=2^{8}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [256, 26347]
Character table: Data not available.