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Properties

Label 16T675
Order \(256\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $(C_2^3\times C_4).D_4$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $675$
Group :  $(C_2^3\times C_4).D_4$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $5$
Generators:  (1,8,2,7)(3,5,4,6)(9,11,10,12)(13,14)(15,16), (1,16,7,11,3,14,6,9,2,15,8,12,4,13,5,10)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_4$ x 2, $C_2^2$
8:  $D_{4}$ x 2, $C_4\times C_2$
16:  $D_{8}$, $QD_{16}$, $C_2^2:C_4$
32:  $C_4\wr C_2$, $C_2^3 : C_4 $, 16T26
64:  $((C_8 : C_2):C_2):C_2$, $(((C_4 \times C_2): C_2):C_2):C_2$, 16T163
128:  16T330

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $D_{8}$

Low degree siblings

16T675 x 3, 32T3277 x 2, 32T3278 x 4, 32T5767 x 4, 32T7554 x 2

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

Group invariants

Order:  $256=2^{8}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [256, 384]
Character table: Data not available.