# Properties

 Label 10T5 Degree $10$ Order $40$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $F_{5}\times C_2$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(10, 5);

## Group action invariants

 Degree $n$: $10$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $5$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $F_{5}\times C_2$ CHM label: $F(5)[x]2$ 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: (1,7,9,3)(2,4,8,6), (1,2,3,4,5,6,7,8,9,10) 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 5: $F_5$

## Low degree siblings

10T5, 20T9, 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 Index Representative 1A $1^{10}$ $1$ $1$ $0$ $()$ 2A $2^{5}$ $1$ $2$ $5$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$ 2B $2^{5}$ $5$ $2$ $5$ $( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,10)$ 2C $2^{4},1^{2}$ $5$ $2$ $4$ $( 2,10)( 3, 9)( 4, 8)( 5, 7)$ 4A1 $4^{2},1^{2}$ $5$ $4$ $6$ $( 2, 4,10, 8)( 3, 7, 9, 5)$ 4A-1 $4^{2},2$ $5$ $4$ $7$ $( 1,10, 7, 8)( 2, 3, 6, 5)( 4, 9)$ 4B1 $4^{2},2$ $5$ $4$ $7$ $( 1, 2, 9, 8)( 3, 6, 7, 4)( 5,10)$ 4B-1 $4^{2},1^{2}$ $5$ $4$ $6$ $( 2, 8,10, 4)( 3, 5, 9, 7)$ 5A $5^{2}$ $4$ $5$ $8$ $( 1, 7, 3, 9, 5)( 2, 8, 4,10, 6)$ 10A $10$ $4$ $10$ $9$ $( 1, 4, 7,10, 3, 6, 9, 2, 5, 8)$

magma: ConjugacyClasses(G);

Malle's constant $a(G)$:     $1/4$

## 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 4B1 4A1 4A-1 4B-1 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);