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Properties

Label	20T9
Degree	$20$
Order	$40$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_2\times F_5$

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Show commands: Magma

magma: G := TransitiveGroup(20, 9);

Group action invariants

Degree $n$:		$20$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:		$9$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:		$C_2\times F_5$
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:		(1,17,13,9,6,2,18,14,10,5)(3,20,16,11,8,4,19,15,12,7), (1,15,17,3)(2,16,18,4)(5,8,13,11)(6,7,14,12)(9,19,10,20)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$ x 3
$4$: $C_4$ x 2, $C_2^2$
$8$: $C_4\times C_2$
$20$: $F_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$
Degree 4: $C_4$
Degree 5: $F_5$
Degree 10: $F_5$

Low degree siblings

10T5 x 2, 20T13, 40T14
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, 1, 1, 1, 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, 1, 1, 1 $	$5$	$2$	$( 3,19)( 4,20)( 5,17)( 6,18)( 7,15)( 8,16)( 9,14)(10,13)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$5$	$2$	$( 1, 2)( 3,20)( 4,19)( 5,18)( 6,17)( 7,16)( 8,15)( 9,13)(10,14)(11,12)$
	$ 4, 4, 4, 4, 4 $	$5$	$4$	$( 1, 3, 9, 7)( 2, 4,10, 8)( 5,15, 6,16)(11,13,19,17)(12,14,20,18)$
	$ 4, 4, 4, 4, 4 $	$5$	$4$	$( 1, 3,17,15)( 2, 4,18,16)( 5,11,13, 8)( 6,12,14, 7)( 9,20,10,19)$
	$ 4, 4, 4, 4, 4 $	$5$	$4$	$( 1, 4, 9, 8)( 2, 3,10, 7)( 5,16, 6,15)(11,14,19,18)(12,13,20,17)$
	$ 4, 4, 4, 4, 4 $	$5$	$4$	$( 1, 4,17,16)( 2, 3,18,15)( 5,12,13, 7)( 6,11,14, 8)( 9,19,10,20)$
	$ 10, 10 $	$4$	$10$	$( 1, 5,10,14,18, 2, 6, 9,13,17)( 3, 7,12,15,19, 4, 8,11,16,20)$
	$ 5, 5, 5, 5 $	$4$	$5$	$( 1, 6,10,13,18)( 2, 5, 9,14,17)( 3, 8,12,16,19)( 4, 7,11,15,20)$

magma: ConjugacyClasses(G);

Group invariants

Order:		$40=2^{3} \cdot 5$	magma: Order(G);
Cyclic:		no	magma: IsCyclic(G);
Abelian:		no	magma: IsAbelian(G);
Solvable:		yes	magma: IsSolvable(G);
Nilpotency class:		not nilpotent
Label:		40.12	magma: IdentifyGroup(G);
Character table:

		1A	2A	2B	2C	4A1	4A-1	4B1	4B-1	5A	10A
	Size	1	1	5	5	5	5	5	5	4	4
	2 P	1A	1A	1A	1A	2C	2C	2C	2C	5A	5A
	5 P	1A	2A	2B	2C	4A-1	4B-1	4B1	4A1	1A	2A
	Type
40.12.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
40.12.1b	R	$1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$
40.12.1c	R	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$
40.12.1d	R	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$
40.12.1e1	C	$1$	$- 1$	$1$	$- 1$	$- i$	$i$	$i$	$- i$	$1$	$- 1$
40.12.1e2	C	$1$	$- 1$	$1$	$- 1$	$i$	$- i$	$- i$	$i$	$1$	$- 1$
40.12.1f1	C	$1$	$1$	$- 1$	$- 1$	$- i$	$i$	$- i$	$i$	$1$	$1$
40.12.1f2	C	$1$	$1$	$- 1$	$- 1$	$i$	$- i$	$i$	$- i$	$1$	$1$
40.12.4a	R	$4$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$	$- 1$
40.12.4b	R	$4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$	$1$

magma: CharacterTable(G);