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Properties

Label 16T268
Order \(128\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^4.C_2^3$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $268$
Group :  $C_2^4.C_2^3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,16,4,13)(2,15,3,14)(5,9,7,12)(6,10,8,11), (1,5,2,6)(3,8,4,7)(9,16,10,15)(11,14,12,13), (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 7
4:  $C_4$ x 4, $C_2^2$ x 7
8:  $D_{4}$ x 8, $C_4\times C_2$ x 6, $C_2^3$
16:  $D_4\times C_2$ x 4, $C_2^2:C_4$ x 4, $Q_8:C_2$ x 2, $C_4\times C_2^2$
32:  $C_2^2 \wr C_2$, $C_4 \times D_4$ x 2, $C_2 \times (C_2^2:C_4)$, 16T34 x 2, 16T37
64:  32T239

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

16T218 x 2, 16T224 x 2, 16T284, 16T295, 16T304, 16T318 x 2, 16T326 x 2, 32T480 x 2, 32T481 x 4, 32T497 x 2, 32T498 x 2, 32T499 x 2, 32T622 x 2, 32T623, 32T658, 32T659, 32T684 x 2, 32T685, 32T686 x 2, 32T732 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, 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, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $2$ $( 3, 4)( 7, 8)(11,12)(15,16)$
$ 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 $ $2$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,15)(14,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,15,10,16)(11,14,12,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,15)(10,16)(11,13)(12,14)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 5, 2, 6)( 3, 8, 4, 7)( 9,15,10,16)(11,13,12,14)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,13,10,14)(11,16,12,15)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 7, 2, 8)( 3, 6, 4, 5)( 9,13,10,14)(11,15,12,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,13)(10,14)(11,15)(12,16)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 9, 3,11)( 2,10, 4,12)( 5,15, 8,13)( 6,16, 7,14)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 9, 4,12)( 2,10, 3,11)( 5,15, 7,14)( 6,16, 8,13)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 9, 3,12)( 2,10, 4,11)( 5,15, 7,13)( 6,16, 8,14)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1,11, 3, 9)( 2,12, 4,10)( 5,13, 8,15)( 6,14, 7,16)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1,11, 4,10)( 2,12, 3, 9)( 5,13, 7,16)( 6,14, 8,15)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1,11, 4, 9)( 2,12, 3,10)( 5,13, 8,16)( 6,14, 7,15)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1,13, 3,15)( 2,14, 4,16)( 5,11, 8, 9)( 6,12, 7,10)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1,13, 4,16)( 2,14, 3,15)( 5,11, 7,10)( 6,12, 8, 9)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1,13, 3,16)( 2,14, 4,15)( 5,11, 7, 9)( 6,12, 8,10)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1,15, 4,14)( 2,16, 3,13)( 5, 9, 7,12)( 6,10, 8,11)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1,15, 3,13)( 2,16, 4,14)( 5, 9, 8,11)( 6,10, 7,12)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1,15, 3,14)( 2,16, 4,13)( 5, 9, 7,11)( 6,10, 8,12)$

Group invariants

Order:  $128=2^{7}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [128, 645]
Character table: Data not available.