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Properties

Label	10T8

Degree	$10$
Order	$80$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$ -group	no
Group:	$C_2^4 : C_5$

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magma:G := TransitiveGroup(10, 8);

Group invariants

Abstract group:		$C_2^4 : C_5$	magma:IdentifyGroup(G);
Order:		$80=2^{4} \cdot 5$	magma:Order(G);
Cyclic:		no	magma:IsCyclic(G);
Abelian:		no	magma:IsAbelian(G);
Solvable:		yes	magma:IsSolvable(G);
Nilpotency class:		not nilpotent	magma:NilpotencyClass(G);

Group action invariants

Degree $n$ :		$10$	magma:t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$ :		$8$	magma:t, n := TransitiveGroupIdentification(G); t;
CHM label:		$[2^{4}]5$
Parity:		$1$	magma:IsEven(G);
Primitive:		no	magma:IsPrimitive(G);
$\card{\Aut(F/K)}$ :		$2$	magma:Order(Centralizer(SymmetricGroup(n), G));
Generators:		$(1,3,5,7,9)(2,4,6,8,10)$ , $(2,7)(5,10)$	magma:Generators(G);

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$5$ : $C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None
Degree 5: $C_5$

Low degree siblings

10T8 x 2, 16T178, 20T17 x 6, 20T23, 40T57 x 3
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	Index	Representative
1A	$1^{10}$	$1$	$1$	$0$	$()$
2A	$2^{2},1^{6}$	$5$	$2$	$2$	$(3,8)(4,9)$
2B	$2^{4},1^{2}$	$5$	$2$	$4$	$( 1, 6)( 3, 8)( 4, 9)( 5,10)$
2C	$2^{2},1^{6}$	$5$	$2$	$2$	$(2,7)(4,9)$
5A1	$5^{2}$	$16$	$5$	$8$	$( 1, 7, 3, 9, 5)( 2, 8, 4,10, 6)$
5A-1	$5^{2}$	$16$	$5$	$8$	$( 1, 5, 9, 3, 7)( 2, 6,10, 4, 8)$
5A2	$5^{2}$	$16$	$5$	$8$	$( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$
5A-2	$5^{2}$	$16$	$5$	$8$	$( 1, 9, 7, 5, 3)( 2,10, 8, 6, 4)$

Malle's constant $a(G)$ : $1/2$

magma:ConjugacyClasses(G);

Character table

		1A	2A	2B	2C	5A1	5A-1	5A2	5A-2
	Size	1	5	5	5	16	16	16	16
	2 P	1A	1A	1A	1A	5A2	5A-2	5A-1	5A1
	5 P	1A	2A	2B	2C	1A	1A	1A	1A
	Type
80.49.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
80.49.1b1	C	$1$	$1$	$1$	$1$	$ζ_{5}^{- 2}$	$ζ_{5}^{2}$	$ζ_{5}$	$ζ_{5}^{- 1}$
80.49.1b2	C	$1$	$1$	$1$	$1$	$ζ_{5}^{2}$	$ζ_{5}^{- 2}$	$ζ_{5}^{- 1}$	$ζ_{5}$
80.49.1b3	C	$1$	$1$	$1$	$1$	$ζ_{5}^{- 1}$	$ζ_{5}$	$ζ_{5}^{- 2}$	$ζ_{5}^{2}$
80.49.1b4	C	$1$	$1$	$1$	$1$	$ζ_{5}$	$ζ_{5}^{- 1}$	$ζ_{5}^{2}$	$ζ_{5}^{- 2}$
80.49.5a	R	$5$	$- 3$	$1$	$1$	$0$	$0$	$0$	$0$
80.49.5b	R	$5$	$1$	$- 3$	$1$	$0$	$0$	$0$	$0$
80.49.5c	R	$5$	$1$	$1$	$- 3$	$0$	$0$	$0$	$0$

magma:CharacterTable(G);

Regular extensions

$f_{ 1 } =$	$16 x^{10} + \left(4 t^{2} - 20\right) x^{8} + \left(-24 t^{2} - 40\right) x^{6} + \left(-t^{4} + 14 t^{2} + 35\right) x^{4} + \left(2 t^{4} + 16 t^{2} + 30\right) x^{2} + \left(-t^{4} - 6 t^{2} - 9\right)$ 16x^10 + (4t^2 - 20)x^8 + (-24t^2 - 40)x^6 + (-t^4 + 14t^2 + 35)x^4 + (2t^4 + 16t^2 + 30)x^2 + (-t^4 - 6*t^2 - 9)