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Properties

Label 16T1243
Degree $16$
Order $1024$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $(C_2^2\times C_4^2).\OD_{16}$

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Show commands: Magma

magma: G := TransitiveGroup(16, 1243);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1243$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $(C_2^2\times C_4^2).\OD_{16}$
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:  (1,16,10,7,3,14,12,6)(2,15,9,8,4,13,11,5), (1,4,2,3)(5,7)(6,8)(13,14)
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, $C_8$ x 2, $C_4\times C_2$
$16$:  $C_8:C_2$, $C_2^2:C_4$, $C_8\times C_2$
$32$:  $(C_8:C_2):C_2$, $C_2^3 : C_4 $, $C_2^2 : C_8$
$64$:  16T84, 16T140 x 2
$128$:  32T1650
$256$:  16T495
$512$:  32T26977

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 8: $(C_8:C_2):C_2$

Low degree siblings

16T1138 x 4, 16T1243 x 3, 32T35730 x 4, 32T35731 x 4, 32T35732 x 2, 32T35733 x 4, 32T35734 x 2, 32T36573 x 4, 32T36574 x 4, 32T36575 x 2, 32T36576 x 2, 32T47040 x 2, 32T47044 x 2, 32T50822 x 2, 32T50824 x 2, 32T71017 x 2, 32T88062 x 2, 32T89296 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 $ $2$ $2$ $( 5, 6)( 7, 8)(13,14)(15,16)$
$ 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 $ $16$ $4$ $( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)$
$ 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 $ $8$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,15)(14,16)$
$ 2, 2, 2, 2, 2, 2, 2, 2 $ $8$ $2$ $( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $32$ $2$ $( 3, 4)( 5, 8)( 6, 7)( 9,10)(13,15)(14,16)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $8$ $4$ $( 5, 7, 6, 8)( 9,10)(11,12)(13,16,14,15)$
$ 4, 4, 2, 2, 1, 1, 1, 1 $ $8$ $4$ $( 1, 2)( 3, 4)( 5, 8, 6, 7)(13,15,14,16)$
$ 4, 4, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $4$ $( 5, 7, 6, 8)(13,15,14,16)$
$ 4, 4, 2, 2, 2, 2 $ $8$ $4$ $( 1, 2)( 3, 4)( 5, 8, 6, 7)( 9,10)(11,12)(13,16,14,15)$
$ 4, 4, 4, 4 $ $32$ $4$ $( 1,10, 3,12)( 2, 9, 4,11)( 5,15, 8,13)( 6,16, 7,14)$
$ 4, 4, 4, 4 $ $32$ $4$ $( 1,10, 4,12)( 2, 9, 3,11)( 5,16, 8,13)( 6,15, 7,14)$
$ 4, 4, 4, 4 $ $32$ $4$ $( 1, 9, 3,12)( 2,10, 4,11)( 5,13, 8,15)( 6,14, 7,16)$
$ 4, 4, 4, 4 $ $32$ $4$ $( 1, 9, 4,12)( 2,10, 3,11)( 5,13, 7,15)( 6,14, 8,16)$
$ 8, 8 $ $64$ $8$ $( 1,16,10, 7, 3,14,12, 6)( 2,15, 9, 8, 4,13,11, 5)$
$ 8, 8 $ $64$ $8$ $( 1,15, 9, 7, 4,13,12, 6)( 2,16,10, 8, 3,14,11, 5)$
$ 8, 8 $ $64$ $8$ $( 1, 7,12,16, 3, 6,10,14)( 2, 8,11,15, 4, 5, 9,13)$
$ 8, 8 $ $64$ $8$ $( 1, 8,12,15, 3, 6,10,14)( 2, 7,11,16, 4, 5, 9,13)$
$ 4, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $16$ $4$ $( 7, 8)( 9,12,10,11)(13,15)(14,16)$
$ 4, 2, 2, 2, 2, 2, 1, 1 $ $16$ $4$ $( 1, 2)( 3, 4)( 5, 6)( 9,11,10,12)(13,16)(14,15)$
$ 4, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $16$ $4$ $( 7, 8)( 9,12,10,11)(13,16)(14,15)$
$ 4, 2, 2, 2, 2, 2, 1, 1 $ $16$ $4$ $( 1, 2)( 3, 4)( 5, 6)( 9,11,10,12)(13,15)(14,16)$
$ 4, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $16$ $4$ $( 3, 4)( 9,11)(10,12)(13,16,14,15)$
$ 4, 2, 2, 2, 2, 2, 1, 1 $ $16$ $4$ $( 1, 2)( 5, 6)( 7, 8)( 9,12)(10,11)(13,15,14,16)$
$ 4, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ $16$ $4$ $( 3, 4)( 9,11)(10,12)(13,15,14,16)$
$ 4, 2, 2, 2, 2, 2, 1, 1 $ $16$ $4$ $( 1, 2)( 5, 6)( 7, 8)( 9,12)(10,11)(13,16,14,15)$
$ 4, 4, 2, 2, 2, 2 $ $64$ $4$ $( 1,10, 2, 9)( 3,12)( 4,11)( 5,15)( 6,16)( 7,13, 8,14)$
$ 4, 4, 2, 2, 2, 2 $ $64$ $4$ $( 1,10, 2, 9)( 3,11)( 4,12)( 5,16, 6,15)( 7,13)( 8,14)$
$ 8, 8 $ $64$ $8$ $( 1,16,12, 7, 4,13, 9, 6)( 2,15,11, 8, 3,14,10, 5)$
$ 8, 8 $ $64$ $8$ $( 1,15,12, 8, 4,13, 9, 6)( 2,16,11, 7, 3,14,10, 5)$
$ 8, 8 $ $64$ $8$ $( 1, 7,11,13, 4, 5, 9,16)( 2, 8,12,14, 3, 6,10,15)$
$ 8, 8 $ $64$ $8$ $( 1, 8,11,13, 3, 6,10,16)( 2, 7,12,14, 4, 5, 9,15)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $1024=2^{10}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:  $7$
Label:  1024.det
magma: IdentifyGroup(G);
 
Character table:    37 x 37 character table

magma: CharacterTable(G);