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Properties

Label	16T27
Degree	$16$
Order	$32$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	yes
Group:	$C_4^2:C_2$

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Show commands: Magma

magma: G := TransitiveGroup(16, 27);

Group action invariants

Degree $n$:		$16$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:		$27$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:		$C_4^2:C_2$
Parity:		$1$	magma: IsEven(G);
Primitive:		no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:		$4$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:		(1,5,9,14)(2,6,10,13)(3,16,12,7)(4,15,11,8), (1,2)(3,12)(4,11)(7,15)(8,16)(9,10), (1,4,2,3)(5,8,6,7)(9,11,10,12)(13,16,14,15)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$ x 7
$4$: $C_2^2$ x 7
$8$: $C_2^3$
$16$: $Q_8:C_2$ x 3

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3
Degree 4: $C_2^2$
Degree 8: $Q_8:C_2$ x 3

Low degree siblings

32T13
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
	$1^{16}$	$1$	$1$	$()$
	$2^{6},1^{4}$	$4$	$2$	$( 3,11)( 4,12)( 5, 6)( 7,16)( 8,15)(13,14)$
	$2^{8}$	$1$	$2$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)$
	$4^{4}$	$4$	$4$	$( 1, 3, 2, 4)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)$
	$4^{4}$	$2$	$4$	$( 1, 3, 9,12)( 2, 4,10,11)( 5, 8,14,15)( 6, 7,13,16)$
	$4^{4}$	$2$	$4$	$( 1, 4, 9,11)( 2, 3,10,12)( 5, 7,14,16)( 6, 8,13,15)$
	$4^{4}$	$2$	$4$	$( 1, 5,10,13)( 2, 6, 9,14)( 3, 8,11,16)( 4, 7,12,15)$
	$4^{4}$	$4$	$4$	$( 1, 5, 9,14)( 2, 6,10,13)( 3,16,12, 7)( 4,15,11, 8)$
	$4^{4}$	$4$	$4$	$( 1, 7,10,15)( 2, 8, 9,16)( 3, 5,11,13)( 4, 6,12,14)$
	$4^{4}$	$2$	$4$	$( 1, 7, 2, 8)( 3,13, 4,14)( 5,12, 6,11)( 9,16,10,15)$
	$4^{4}$	$2$	$4$	$( 1, 8, 2, 7)( 3,14, 4,13)( 5,11, 6,12)( 9,15,10,16)$
	$2^{8}$	$1$	$2$	$( 1, 9)( 2,10)( 3,12)( 4,11)( 5,14)( 6,13)( 7,16)( 8,15)$
	$2^{8}$	$1$	$2$	$( 1,10)( 2, 9)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16)$
	$4^{4}$	$2$	$4$	$( 1,13,10, 5)( 2,14, 9, 6)( 3,16,11, 8)( 4,15,12, 7)$

magma: ConjugacyClasses(G);

Group invariants

Order:		$32=2^{5}$	magma: Order(G);
Cyclic:		no	magma: IsCyclic(G);
Abelian:		no	magma: IsAbelian(G);
Solvable:		yes	magma: IsSolvable(G);
Nilpotency class:		$2$
Label:		32.33	magma: IdentifyGroup(G);
Character table:

		1A	2A	2B	2C	2D	4A1	4A-1	4B1	4B-1	4C1	4C-1	4D	4E	4F
	Size	1	1	1	1	4	2	2	2	2	2	2	4	4	4
	2 P	1A	1A	1A	1A	1A	2B	2B	2C	2C	2A	2A	2B	2A	2C
	Type
32.33.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
32.33.1b	R	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$1$	$1$
32.33.1c	R	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$
32.33.1d	R	$1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- 1$
32.33.1e	R	$1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$
32.33.1f	R	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$
32.33.1g	R	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$- 1$
32.33.1h	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$
32.33.2a1	C	$2$	$- 2$	$- 2$	$2$	$0$	$0$	$0$	$- 2 i$	$2 i$	$0$	$0$	$0$	$0$	$0$
32.33.2a2	C	$2$	$- 2$	$- 2$	$2$	$0$	$0$	$0$	$2 i$	$- 2 i$	$0$	$0$	$0$	$0$	$0$
32.33.2b1	C	$2$	$- 2$	$2$	$- 2$	$0$	$- 2 i$	$2 i$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
32.33.2b2	C	$2$	$- 2$	$2$	$- 2$	$0$	$2 i$	$- 2 i$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
32.33.2c1	C	$2$	$2$	$- 2$	$- 2$	$0$	$0$	$0$	$0$	$0$	$- 2 i$	$2 i$	$0$	$0$	$0$
32.33.2c2	C	$2$	$2$	$- 2$	$- 2$	$0$	$0$	$0$	$0$	$0$	$2 i$	$- 2 i$	$0$	$0$	$0$

magma: CharacterTable(G);