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Properties

Label	16T6
Degree	$16$
Order	$16$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	yes
Group:	$C_8: C_2$

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Show commands: Magma

magma: G := TransitiveGroup(16, 6);

Group action invariants

Degree $n$:		$16$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:		$6$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:		$C_8: C_2$
Parity:		$1$	magma: IsEven(G);
Primitive:		no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:		$16$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:		(1,9)(2,10)(3,4)(5,13)(6,14)(7,8)(11,12)(15,16), (1,3,5,8,10,12,14,15)(2,4,6,7,9,11,13,16)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$ x 3
$4$: $C_4$ x 2, $C_2^2$
$8$: $C_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3
Degree 4: $C_4$ x 2, $C_2^2$
Degree 8: $C_4\times C_2$, $C_8:C_2$

Low degree siblings

8T7
Siblings are shown with degree $\leq 47$

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

Conjugacy classes

Label	Cycle Type	Size	Order	Representative
	$1^{16}$	$1$	$1$	$()$
	$2^{8}$	$2$	$2$	$( 1, 2)( 3,11)( 4,12)( 5, 6)( 7,15)( 8,16)( 9,10)(13,14)$
	$8^{2}$	$2$	$8$	$( 1, 3, 5, 8,10,12,14,15)( 2, 4, 6, 7, 9,11,13,16)$
	$8^{2}$	$2$	$8$	$( 1, 4,14,16,10,11, 5, 7)( 2, 3,13,15, 9,12, 6, 8)$
	$4^{4}$	$1$	$4$	$( 1, 5,10,14)( 2, 6, 9,13)( 3, 8,12,15)( 4, 7,11,16)$
	$4^{4}$	$2$	$4$	$( 1, 6,10,13)( 2, 5, 9,14)( 3,16,12, 7)( 4,15,11, 8)$
	$8^{2}$	$2$	$8$	$( 1, 7, 5,11,10,16,14, 4)( 2, 8, 6,12, 9,15,13, 3)$
	$8^{2}$	$2$	$8$	$( 1, 8,14, 3,10,15, 5,12)( 2, 7,13, 4, 9,16, 6,11)$
	$2^{8}$	$1$	$2$	$( 1,10)( 2, 9)( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)$
	$4^{4}$	$1$	$4$	$( 1,14,10, 5)( 2,13, 9, 6)( 3,15,12, 8)( 4,16,11, 7)$

magma: ConjugacyClasses(G);

Group invariants

Order:		$16=2^{4}$	magma: Order(G);
Cyclic:		no	magma: IsCyclic(G);
Abelian:		no	magma: IsAbelian(G);
Solvable:		yes	magma: IsSolvable(G);
Nilpotency class:		$2$
Label:		16.6	magma: IdentifyGroup(G);
Character table:

		1A	2A	2B	4A1	4A-1	4B	8A1	8A-1	8B1	8B-1
	Size	1	1	2	1	1	2	2	2	2	2
	2 P	1A	1A	1A	2A	2A	2A	4A1	4A-1	4A1	4A-1
	Type
16.6.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
16.6.1b	R	$1$	$1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$
16.6.1c	R	$1$	$1$	$- 1$	$1$	$1$	$- 1$	$1$	$1$	$- 1$	$- 1$
16.6.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$
16.6.1e1	C	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$- i$	$i$	$- i$	$i$
16.6.1e2	C	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$i$	$- i$	$i$	$- i$
16.6.1f1	C	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- i$	$i$	$i$	$- i$
16.6.1f2	C	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$i$	$- i$	$- i$	$i$
16.6.2a1	C	$2$	$- 2$	$0$	$- 2 i$	$2 i$	$0$	$0$	$0$	$0$	$0$
16.6.2a2	C	$2$	$- 2$	$0$	$2 i$	$- 2 i$	$0$	$0$	$0$	$0$	$0$

magma: CharacterTable(G);