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Properties

Label	20T13
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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magma: G := TransitiveGroup(20, 13);

Group action invariants

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

Low degree siblings

10T5 x 2, 20T9, 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 $	$5$	$2$	$( 3,19)( 4,20)( 5,18)( 6,17)( 7,15)( 8,16)( 9,13)(10,14)$
	$ 4, 4, 4, 4, 2, 2 $	$5$	$4$	$( 1, 2)( 3, 7,19,15)( 4, 8,20,16)( 5,13,18, 9)( 6,14,17,10)(11,12)$
	$ 4, 4, 4, 4, 2, 2 $	$5$	$4$	$( 1, 2)( 3,15,19, 7)( 4,16,20, 8)( 5, 9,18,13)( 6,10,17,14)(11,12)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$5$	$2$	$( 1, 3)( 2, 4)( 5,19)( 6,20)( 7,17)( 8,18)( 9,15)(10,16)(11,13)(12,14)$
	$ 10, 10 $	$4$	$10$	$( 1, 3, 5, 8,10,12,14,16,18,19)( 2, 4, 6, 7, 9,11,13,15,17,20)$
	$ 4, 4, 4, 4, 2, 2 $	$5$	$4$	$( 1, 4,10, 7)( 2, 3, 9, 8)( 5,15)( 6,16)(11,14,20,18)(12,13,19,17)$
	$ 4, 4, 4, 4, 2, 2 $	$5$	$4$	$( 1, 4,18,15)( 2, 3,17,16)( 5,11,14, 7)( 6,12,13, 8)( 9,19)(10,20)$
	$ 5, 5, 5, 5 $	$4$	$5$	$( 1, 5,10,14,18)( 2, 6, 9,13,17)( 3, 8,12,16,19)( 4, 7,11,15,20)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1,12)( 2,11)( 3,14)( 4,13)( 5,16)( 6,15)( 7,17)( 8,18)( 9,20)(10,19)$

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);