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Properties

Label 16T364
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 $t$ :  $364$
Group :  $C_2^4.C_2^3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,6)(2,5)(3,15,11,7)(4,16,12,8)(9,13)(10,14), (1,12,9,4)(2,11,10,3)(5,8,14,16)(6,7,13,15), (1,2)(3,4)(5,6)(7,16)(8,15)(9,10)(11,12)(13,14)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
8:  $D_{4}$ x 12, $C_2^3$
16:  $D_4\times C_2$ x 6, $Q_8:C_2$
32:  $C_2^2 \wr C_2$ x 3, 16T34 x 3, $C_4^2:C_2$
64:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 2, 32T320

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 8: $D_4$, $(((C_4 \times C_2): C_2):C_2):C_2$ x 2

Low degree siblings

16T350 x 4, 16T364 x 3, 16T373 x 8, 16T392 x 16, 32T801 x 8, 32T802 x 4, 32T803 x 2, 32T804, 32T830, 32T831 x 4, 32T832 x 8, 32T845 x 4, 32T846 x 2, 32T878 x 4, 32T879 x 4, 32T1554, 32T1709 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 $ $4$ $2$ $( 7,15)( 8,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 5,14)( 6,13)( 7,15)( 8,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 3,11)( 4,12)( 7,15)( 8,16)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 3,11)( 4,12)( 5,14)( 6,13)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $4$ $2$ $( 3,11)( 4,12)( 5,14)( 6,13)( 7,15)( 8,16)$
$ 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 $ $4$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7,16)( 8,15)( 9,10)(11,12)(13,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 2)( 3, 4)( 5,13)( 6,14)( 7,16)( 8,15)( 9,10)(11,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 2)( 3,12)( 4,11)( 5, 6)( 7,16)( 8,15)( 9,10)(13,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 2)( 3,12)( 4,11)( 5,13)( 6,14)( 7, 8)( 9,10)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 2)( 3,12)( 4,11)( 5,13)( 6,14)( 7,16)( 8,15)( 9,10)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,11)(10,12)(13,16)(14,15)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 3)( 2, 4)( 5, 7,14,15)( 6, 8,13,16)( 9,11)(10,12)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 3, 9,11)( 2, 4,10,12)( 5, 7)( 6, 8)(13,16)(14,15)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 3, 9,11)( 2, 4,10,12)( 5, 7,14,15)( 6, 8,13,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 5)( 2, 6)( 3, 8)( 4, 7)( 9,14)(10,13)(11,16)(12,15)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 5)( 2, 6)( 3, 8,11,16)( 4, 7,12,15)( 9,14)(10,13)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 5, 9,14)( 2, 6,10,13)( 3, 8)( 4, 7)(11,16)(12,15)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 5, 9,14)( 2, 6,10,13)( 3, 8,11,16)( 4, 7,12,15)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 7, 2, 8)( 3, 6, 4, 5)( 9,15,10,16)(11,13,12,14)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 7,10,16)( 2, 8, 9,15)( 3, 6, 4, 5)(11,13,12,14)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 7, 2, 8)( 3, 6,12,14)( 4, 5,11,13)( 9,15,10,16)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 7,10,16)( 2, 8, 9,15)( 3, 6,12,14)( 4, 5,11,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 9)( 2,10)( 3,11)( 4,12)( 5,14)( 6,13)( 7,15)( 8,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1,10)( 2, 9)( 3,12)( 4,11)( 5,13)( 6,14)( 7,16)( 8,15)$

Group invariants

Order:  $128=2^{7}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [128, 753]
Character table: Data not available.