↑

Properties

Label 16T392
Order \(128\)
n \(16\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group Yes
Group: $C_2^4.C_2^3$

Related objects

Learn more about

Group action invariants

Degree $n$ :  $16$
Transitive number $t$ :  $392$
Group :  $C_2^4.C_2^3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,14,2,13)(3,11,4,12)(5,9,6,10)(7,15,8,16), (1,4,6,15)(2,3,5,16)(9,13)(10,14), (1,10)(2,9)(3,11)(4,12)(5,13)(6,14)(7,16)(8,15)
$|\Aut(F/K)|$:  $4$

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: $C_2$

Degree 4: $D_{4}$ x 3

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

Low degree siblings

16T350 x 4, 16T364 x 4, 16T373 x 8, 16T392 x 15, 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, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 7,10)( 8, 9)(11,14)(12,13)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 7,11)( 8,12)( 9,13)(10,14)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $8$ $2$ $( 3,16)( 4,15)( 7, 8)( 9,14)(10,13)(11,12)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $8$ $4$ $( 3,16)( 4,15)( 7, 9,11,13)( 8,10,12,14)$
$ 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 $ $4$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 9)( 8,10)(11,13)(12,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7,12)( 8,11)( 9,14)(10,13)(15,16)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 2)( 3,15)( 4,16)( 5, 6)( 7,10,11,14)( 8, 9,12,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 3)( 2, 4)( 5,15)( 6,16)( 7,10)( 8, 9)(11,14)(12,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 3)( 2, 4)( 5,15)( 6,16)( 7,11)( 8,12)( 9,13)(10,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 3)( 2, 4)( 5,15)( 6,16)( 7,14)( 8,13)( 9,12)(10,11)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 3, 6,16)( 2, 4, 5,15)( 7, 9,11,13)( 8,10,12,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 4)( 2, 3)( 5,16)( 6,15)( 7, 9)( 8,10)(11,13)(12,14)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 4)( 2, 3)( 5,16)( 6,15)( 7,12)( 8,11)( 9,14)(10,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $2$ $2$ $( 1, 4)( 2, 3)( 5,16)( 6,15)( 7,13)( 8,14)( 9,11)(10,12)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 5)( 2, 6)( 3,15)( 4,16)( 7,12)( 8,11)( 9,14)(10,13)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 6)( 2, 5)( 3,16)( 4,15)( 7,11)( 8,12)( 9,13)(10,14)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 7, 2, 8)( 3,10, 4, 9)( 5,12, 6,11)(13,16,14,15)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 7, 4, 9)( 2, 8, 3,10)( 5,12,16,14)( 6,11,15,13)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 7, 5,12)( 2, 8, 6,11)( 3,10,15,13)( 4, 9,16,14)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 7,15,13)( 2, 8,16,14)( 3,10, 5,12)( 4, 9, 6,11)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 7)( 2, 8)( 3,14)( 4,13)( 5,12)( 6,11)( 9,15)(10,16)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 7, 3,14)( 2, 8, 4,13)( 5,12,15, 9)( 6,11,16,10)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 7, 6,11)( 2, 8, 5,12)( 3,14,16,10)( 4,13,15, 9)$
$ 4, 4, 4, 4 $ $8$ $4$ $( 1, 7,16,10)( 2, 8,15, 9)( 3,14, 6,11)( 4,13, 5,12)$

Group invariants

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