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Properties

Label 16T554
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 $t$ :  $554$
Group :  $C_2^3.C_2^4.C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $4$
Generators:  (1,4,6,7)(2,3,5,8)(9,12,10,11)(13,15,14,16), (1,7,6,4)(2,8,5,3)(9,16,10,15)(11,14,12,13), (1,10,3,12,2,9,4,11)(5,13,7,16,6,14,8,15)
$|\Aut(F/K)|$:  $4$

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 4, $C_4\times C_2$ x 6, $C_2^3$
16:  $C_8:C_2$ x 4, $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $C_4\times C_2^2$
32:  $(C_8:C_2):C_2$ x 2, $C_2^3 : C_4 $ x 2, $C_2 \times (C_8:C_2)$ x 2, $C_2 \times (C_2^2:C_4)$
64:  $((C_8 : C_2):C_2):C_2$ x 4, 16T72, 16T76, 16T95
128:  16T227 x 2, 16T252

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $C_8:C_2$, $((C_8 : C_2):C_2):C_2$ x 2

Low degree siblings

16T485 x 32, 16T554 x 31, 32T2419 x 64, 32T2420 x 64, 32T2421 x 16, 32T2422 x 16, 32T2423 x 8, 32T2424 x 16, 32T2749 x 16, 32T2750 x 8, 32T2751 x 8, 32T2752 x 8, 32T2753 x 8, 32T2754 x 16, 32T6931 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 $ $4$ $2$ $(11,15)(12,16)$
$ 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 $ $4$ $2$ $( 9,10)(11,16)(12,15)(13,14)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 9,13)(10,14)(11,16)(12,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 9,14)(10,13)(11,15)(12,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 3, 7)( 4, 8)(11,15)(12,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $4$ $2$ $( 3, 7)( 4, 8)( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $4$ $2$ $( 3, 7)( 4, 8)( 9,10)(11,16)(12,15)(13,14)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $4$ $2$ $( 3, 7)( 4, 8)( 9,13)(10,14)(11,12)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $4$ $2$ $( 3, 7)( 4, 8)( 9,13)(10,14)(11,16)(12,15)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $4$ $2$ $( 3, 7)( 4, 8)( 9,14)(10,13)(11,15)(12,16)$
$ 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 $ $4$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,16)(12,15)(13,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,13)(10,14)(11,16)(12,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,14)(10,13)(11,15)(12,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 2)( 3, 8)( 4, 7)( 5, 6)( 9,10)(11,16)(12,15)(13,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 2)( 3, 8)( 4, 7)( 5, 6)( 9,13)(10,14)(11,16)(12,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 2)( 3, 8)( 4, 7)( 5, 6)( 9,14)(10,13)(11,15)(12,16)$
$ 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 $ $8$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,13,16)(10,12,14,15)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,13,15)(10,11,14,16)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 3, 6, 8)( 2, 4, 5, 7)( 9,11,13,16)(10,12,14,15)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 3, 6, 8)( 2, 4, 5, 7)( 9,12,10,11)(13,15,14,16)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 3, 6, 8)( 2, 4, 5, 7)( 9,12,13,15)(10,11,14,16)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,12,10,11)(13,15,14,16)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,12,13,15)(10,11,14,16)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 4, 6, 7)( 2, 3, 5, 8)( 9,12,13,15)(10,11,14,16)$
$ 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 $ $1$ $2$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,14)(10,13)(11,15)(12,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,13)(10,14)(11,16)(12,15)$
$ 8, 8 $ $16$ $8$ $( 1, 9, 3,11, 2,10, 4,12)( 5,14, 7,15, 6,13, 8,16)$
$ 8, 8 $ $16$ $8$ $( 1, 9, 3,11, 6,13, 8,16)( 2,10, 4,12, 5,14, 7,15)$
$ 8, 8 $ $16$ $8$ $( 1, 9, 4,12, 2,10, 3,11)( 5,14, 8,16, 6,13, 7,15)$
$ 8, 8 $ $16$ $8$ $( 1, 9, 4,12, 6,13, 7,15)( 2,10, 3,11, 5,14, 8,16)$
$ 8, 8 $ $16$ $8$ $( 1,11, 3,10, 2,12, 4, 9)( 5,15, 7,13, 6,16, 8,14)$
$ 8, 8 $ $16$ $8$ $( 1,11, 7,13, 6,16, 4, 9)( 2,12, 8,14, 5,15, 3,10)$
$ 8, 8 $ $16$ $8$ $( 1,11, 4, 9, 2,12, 3,10)( 5,15, 8,14, 6,16, 7,13)$
$ 8, 8 $ $16$ $8$ $( 1,11, 8,14, 6,16, 3,10)( 2,12, 7,13, 5,15, 4, 9)$

Group invariants

Order:  $256=2^{8}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [256, 4487]
Character table: Data not available.