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Properties

Label 16T75
Order \(64\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $(C_2\times C_8):C_2^2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $75$
Group :  $(C_2\times C_8):C_2^2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,11,2,12)(3,9,4,10)(5,7,6,8)(13,16,14,15), (1,13)(2,14)(3,4)(5,9)(6,10)(7,15)(8,16)(11,12), (1,13)(2,14)(3,11)(4,12)(5,9)(6,10)(7,8)(15,16)
$|\Aut(F/K)|$:  $4$

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:  $D_4\times C_2$ x 2, $C_2^2:C_4$ x 4, $C_4\times C_2^2$
32:  $Z_8 : Z_8^\times$ x 2, $C_2 \times (C_2^2:C_4)$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$, $D_{4}$ x 2

Degree 8: $C_2^2:C_4$, $Z_8 : Z_8^\times$ x 2

Low degree siblings

16T75 x 7, 32T65 x 2, 32T66 x 4

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

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

Group invariants

Order:  $64=2^{6}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [64, 99]
Character table: Data not available.