Properties

 Label 10T4 Degree $10$ Order $20$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $F_5$

Related objects

Show commands: Magma

magma: G := TransitiveGroup(10, 4);

Group action invariants

 Degree $n$: $10$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $4$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $F_5$ CHM label: $1/2[F(5)]2$ Parity: $-1$ magma: IsEven(G); Primitive: no magma: IsPrimitive(G); magma: NilpotencyClass(G); $\card{\Aut(F/K)}$: $2$ magma: Order(Centralizer(SymmetricGroup(n), G)); Generators: (1,3,5,7,9)(2,4,6,8,10), (1,2,9,8)(3,6,7,4)(5,10) magma: Generators(G);

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$4$:  $C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: $F_5$

Low degree siblings

5T3, 20T5

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 1A $1^{10}$ $1$ $1$ $()$ 2A $2^{4},1^{2}$ $5$ $2$ $(1,9)(2,8)(3,7)(4,6)$ 4A1 $4^{2},2$ $5$ $4$ $( 1, 2, 9, 8)( 3, 6, 7, 4)( 5,10)$ 4A-1 $4^{2},2$ $5$ $4$ $( 1, 8, 9, 2)( 3, 4, 7, 6)( 5,10)$ 5A $5^{2}$ $4$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$

magma: ConjugacyClasses(G);

Group invariants

 Order: $20=2^{2} \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Nilpotency class: not nilpotent Label: 20.3 magma: IdentifyGroup(G); Character table:

 1A 2A 4A1 4A-1 5A Size 1 5 5 5 4 2 P 1A 1A 2A 2A 5A 5 P 1A 2A 4A1 4A-1 1A Type 20.3.1a R $1$ $1$ $1$ $1$ $1$ 20.3.1b R $1$ $1$ $−1$ $−1$ $1$ 20.3.1c1 C $1$ $−1$ $−i$ $i$ $1$ 20.3.1c2 C $1$ $−1$ $i$ $−i$ $1$ 20.3.4a R $4$ $0$ $0$ $0$ $−1$

magma: CharacterTable(G);