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Properties

Label 16T1155
Degree $16$
Order $1024$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group yes
Group: $C_2^7.C_8$

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Show commands: Magma

magma: G := TransitiveGroup(16, 1155);
 

Group action invariants

Degree $n$:  $16$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $1155$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_2^7.C_8$
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:  (1,7,9,14,3,5,12,15,2,8,10,13,4,6,11,16), (1,6,9,13,3,8,12,16,2,5,10,14,4,7,11,15)
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 $, $C_2^2 : C_8$
$64$:  $((C_8 : C_2):C_2):C_2$ x 2, 16T84
$128$:  16T228
$256$:  16T565
$512$:  16T817

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

16T1155 x 7, 16T1223 x 8, 32T35849 x 8, 32T35850 x 16, 32T35851 x 8, 32T35852 x 8, 32T35853 x 16, 32T35854 x 4, 32T35855 x 4, 32T36405 x 32, 32T36406 x 8, 32T36407 x 32, 32T36408 x 16, 32T36409 x 8, 32T36410 x 16, 32T36411 x 4, 32T36412 x 16, 32T36413 x 16, 32T36414 x 16, 32T36415 x 16, 32T36416 x 16, 32T36417 x 16, 32T36418 x 8, 32T36419 x 4, 32T36420 x 8, 32T36421 x 16, 32T48622 x 4

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

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.dfi
magma: IdentifyGroup(G);
 
Character table:    40 x 40 character table

magma: CharacterTable(G);