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Properties

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

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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

$\card{(G/N)}$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

LabelCycle TypeSizeOrder IndexRepresentative
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 $10$ $4$ $10$ $9$ $( 1, 4, 7,10, 3, 6, 9, 2, 5, 8)$

Malle's constant $a(G)$:     $1/4$

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 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);