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Properties

Label 16T1441
Degree $16$
Order $2048$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $(C_2\times C_4^3).\SD_{16}$

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Show commands: Magma

magma: G := TransitiveGroup(16, 1441);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1441$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $(C_2\times C_4^3).\SD_{16}$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $2$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,11,4,10)(2,12,3,9)(5,14)(6,13)(7,16,8,15), (1,7,15,10,2,8,16,9)(3,5,13,12)(4,6,14,11)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_4$ x 2, $C_2^2$
$8$:  $D_{4}$ x 2, $C_4\times C_2$
$16$:  $D_{8}$, $QD_{16}$, $C_2^2:C_4$
$32$:  $C_4\wr C_2$, $C_2^3 : C_4 $, 16T26
$64$:  $((C_8 : C_2):C_2):C_2$, $(((C_4 \times C_2): C_2):C_2):C_2$, 16T163
$128$:  16T330
$256$:  16T682
$512$:  16T944
$1024$:  32T55464

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 8: $(((C_4 \times C_2): C_2):C_2):C_2$

Low degree siblings

16T1441 x 7, 16T1448 x 8, 32T99021 x 8, 32T99022 x 4, 32T99023 x 4, 32T99024 x 4, 32T99025 x 4, 32T99026 x 4, 32T99027 x 4, 32T99079 x 16, 32T99080 x 4, 32T99081 x 4, 32T99082 x 16, 32T99083 x 4, 32T99084 x 4, 32T99085 x 4, 32T99086 x 4, 32T116629 x 4, 32T117458 x 4, 32T117463 x 4, 32T145144 x 2, 32T145153 x 2, 32T175918 x 2, 32T182649 x 2, 32T182705 x 2, 32T182722 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 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, 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 $ $4$ $2$ $( 9,10)(11,12)(13,14)(15,16)$
$ 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $2$ $(13,14)(15,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $4$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)$
$ 4, 4, 4, 4 $ $16$ $4$ $( 1, 4, 2, 3)( 5, 8, 6, 7)( 9,12,10,11)(13,16,14,15)$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $4$ $( 5, 8, 6, 7)( 9,12,10,11)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 2)( 3, 4)( 5, 7, 6, 8)( 9,11,10,12)(13,14)(15,16)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $8$ $4$ $( 5, 8, 6, 7)( 9,11,10,12)(13,14)(15,16)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $8$ $4$ $( 1, 2)( 3, 4)( 5, 7, 6, 8)( 9,12,10,11)$
$ 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1 $ $16$ $2$ $( 3, 4)( 7, 8)(11,12)(15,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $16$ $2$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $32$ $2$ $( 3, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(15,16)$
$ 4, 4, 4, 4 $ $128$ $4$ $( 1,15, 3,14)( 2,16, 4,13)( 5,12, 7, 9)( 6,11, 8,10)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $32$ $2$ $( 3, 4)( 7, 8)( 9,12)(10,11)(13,15)(14,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $32$ $2$ $( 3, 4)( 7, 8)( 9,12)(10,11)(13,16)(14,15)$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $16$ $4$ $( 9,11,10,12)(13,16,14,15)$
$ 4, 4, 2, 2, 2, 2 $ $16$ $4$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,12,10,11)(13,15,14,16)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $16$ $4$ $( 5, 6)( 7, 8)( 9,12,10,11)(13,16,14,15)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $16$ $4$ $( 1, 2)( 3, 4)( 9,11,10,12)(13,15,14,16)$
$ 4, 4, 2, 2, 2, 2 $ $64$ $4$ $( 1,15, 2,16)( 3,13)( 4,14)( 5,12)( 6,11)( 7,10, 8, 9)$
$ 4, 4, 2, 2, 2, 2 $ $64$ $4$ $( 1,15, 2,16)( 3,13)( 4,14)( 5,10, 6, 9)( 7,11)( 8,12)$
$ 4, 4, 4, 2, 2 $ $128$ $4$ $( 1,11, 4,10)( 2,12, 3, 9)( 5,14)( 6,13)( 7,16, 8,15)$
$ 4, 4, 4, 2, 2 $ $128$ $4$ $( 1, 9, 2,10)( 3,12)( 4,11)( 5,14, 8,15)( 6,13, 7,16)$
$ 4, 4, 4, 2, 2 $ $128$ $4$ $( 1,10)( 2, 9)( 3,12, 4,11)( 5,16, 7,14)( 6,15, 8,13)$
$ 4, 4, 4, 2, 2 $ $128$ $4$ $( 1,11, 3,10)( 2,12, 4, 9)( 5,16)( 6,15)( 7,14, 8,13)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $32$ $4$ $( 5,10, 6, 9)( 7,11, 8,12)(13,16)(14,15)$
$ 4, 4, 2, 2, 2, 2 $ $32$ $4$ $( 1, 2)( 3, 4)( 5, 9, 6,10)( 7,12, 8,11)(13,15)(14,16)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $32$ $2$ $( 5, 9)( 6,10)( 7,12)( 8,11)(13,15)(14,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $32$ $2$ $( 1, 2)( 3, 4)( 5,10)( 6, 9)( 7,11)( 8,12)(13,16)(14,15)$
$ 4, 4, 4, 2, 1, 1 $ $64$ $4$ $( 1, 4, 2, 3)( 5,11, 6,12)( 7, 9, 8,10)(13,14)$
$ 4, 2, 2, 2, 2, 2, 1, 1 $ $64$ $4$ $( 1, 4, 2, 3)( 5,12)( 6,11)( 7,10)( 8, 9)(15,16)$
$ 8, 2, 2, 2, 1, 1 $ $64$ $8$ $( 3, 4)( 5,11, 7,10, 6,12, 8, 9)(13,14)(15,16)$
$ 8, 2, 1, 1, 1, 1, 1, 1 $ $64$ $8$ $( 1, 2)( 5,12, 7, 9, 6,11, 8,10)$
$ 8, 4, 2, 2 $ $64$ $8$ $( 1, 4)( 2, 3)( 5, 9, 8,12, 6,10, 7,11)(13,15,14,16)$
$ 8, 4, 2, 2 $ $64$ $8$ $( 1, 3)( 2, 4)( 5,10, 8,11, 6, 9, 7,12)(13,16,14,15)$
$ 8, 4, 4 $ $128$ $8$ $( 1,11,15, 5)( 2,12,16, 6)( 3, 9,14, 7, 4,10,13, 8)$
$ 8, 4, 4 $ $128$ $8$ $( 1,12,15, 5)( 2,11,16, 6)( 3, 9,14, 8, 4,10,13, 7)$
$ 8, 4, 4 $ $128$ $8$ $( 1,10,15, 5)( 2, 9,16, 6)( 3,12,14, 8, 4,11,13, 7)$
$ 8, 4, 4 $ $128$ $8$ $( 1,10,16, 6, 2, 9,15, 5)( 3,11,13, 8)( 4,12,14, 7)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $2048=2^{11}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $8$
Label:  2048.cog
magma: IdentifyGroup(G);
 
Character table:    41 x 41 character table

magma: CharacterTable(G);