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Properties

Label 16T317
Order \(128\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^4:C_2^2.C_2$

Related objects

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Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $317$
Group :  $C_2^4:C_2^2.C_2$
Parity:  $1$
Primitive:  No
Nilpotency class:  $4$
Generators:  (1,4,2,3)(5,7,6,8)(9,11,10,12)(13,15,14,16), (1,15,5,9)(2,16,6,10)(3,14,8,12)(4,13,7,11), (1,6,2,5)(3,8,4,7)(9,15)(10,16)(11,14)(12,13)
$|\Aut(F/K)|$:  $2$

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 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:  $C_2^3 : C_4 $ x 2, $C_2 \times (C_2^2:C_4)$
64:  16T76

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

16T219, 16T236, 16T266, 16T287, 16T319 x 2, 16T324, 32T482, 32T483, 32T484, 32T530, 32T531, 32T614, 32T615, 32T616, 32T617, 32T663, 32T664, 32T731 x 2, 32T733 x 2, 32T734 x 2, 32T743

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

Group invariants

Order:  $128=2^{7}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [128, 853]
Character table: Data not available.