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Properties

Label 16T210
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 $t$ :  $210$
Group :  $C_2^4.C_2^3$
Parity:  $1$
Primitive:  No
Nilpotency class:  $3$
Generators:  (1,7,10,16)(2,8,9,15)(3,5)(4,6)(11,14)(12,13), (1,4,9,12)(2,3,10,11)(5,15,14,7)(6,16,13,8), (1,16,9,8)(2,15,10,7)(3,14,11,5)(4,13,12,6)
$|\Aut(F/K)|$:  $4$

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 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:  $(((C_4 \times C_2): C_2):C_2):C_2$ x 2, 32T239

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

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

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

Low degree siblings

16T208 x 4, 16T210 x 3, 16T217 x 8, 16T247 x 8, 32T451 x 2, 32T452, 32T453 x 4, 32T454 x 8, 32T455 x 4, 32T459 x 8, 32T460, 32T478 x 4, 32T479 x 2, 32T564 x 2, 32T565 x 2, 32T1567 x 2, 32T1705

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 $ $2$ $2$ $( 5,13)( 6,14)( 7,16)( 8,15)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $( 3, 4)( 7,15)( 8,16)(11,12)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $4$ $2$ $( 3, 4)( 5,13)( 6,14)( 7, 8)(11,12)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $4$ $2$ $( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16)$
$ 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, 1, 1, 1, 1, 1, 1, 1, 1 $ $2$ $2$ $( 3,12)( 4,11)( 5,13)( 6,14)$
$ 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 $ $2$ $2$ $( 1, 2)( 3, 4)( 5,14)( 6,13)( 7,15)( 8,16)( 9,10)(11,12)$
$ 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 $ $2$ $2$ $( 1, 2)( 3,11)( 4,12)( 5,14)( 6,13)( 7, 8)( 9,10)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 2)( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)( 9,10)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 3)( 2, 4)( 5, 7,13,16)( 6, 8,14,15)( 9,11)(10,12)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 3)( 2, 4)( 5,16,13, 7)( 6,15,14, 8)( 9,11)(10,12)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 3, 2, 4)( 5,16, 6,15)( 7,14, 8,13)( 9,11,10,12)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 3, 9,11)( 2, 4,10,12)( 5, 7,14,15)( 6, 8,13,16)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 3, 9,11)( 2, 4,10,12)( 5,16,14, 8)( 6,15,13, 7)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9,14)(10,13)(11,15)(12,16)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 5,10,13)( 2, 6, 9,14)( 3, 7,12,16)( 4, 8,11,15)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 5)( 2, 6)( 3, 8,12,15)( 4, 7,11,16)( 9,14)(10,13)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 5,10,13)( 2, 6, 9,14)( 3, 8)( 4, 7)(11,16)(12,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $4$ $2$ $( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,13)(10,14)(11,16)(12,15)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 6,10,14)( 2, 5, 9,13)( 3, 8,12,15)( 4, 7,11,16)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 7,10,16)( 2, 8, 9,15)( 3, 5)( 4, 6)(11,14)(12,13)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 7)( 2, 8)( 3, 5,12,13)( 4, 6,11,14)( 9,15)(10,16)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 7, 2, 8)( 3, 6, 4, 5)( 9,15,10,16)(11,13,12,14)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 7, 9,15)( 2, 8,10,16)( 3, 6,11,13)( 4, 5,12,14)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 7, 9,15)( 2, 8,10,16)( 3,14,11, 5)( 4,13,12, 6)$
$ 4, 4, 4, 4 $ $4$ $4$ $( 1, 7, 2, 8)( 3,14, 4,13)( 5,12, 6,11)( 9,15,10,16)$
$ 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, 621]
Character table: Data not available.