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Properties

Label 10T22
Degree $10$
Order $240$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $S_5\times C_2$

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Show commands: Magma

magma: G := TransitiveGroup(10, 22);
 

Group action invariants

Degree $n$:  $10$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $22$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_5\times C_2$
CHM label:   $S(5)[x]2$
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:  (2,10)(5,7), (1,6)(2,7)(3,8)(4,9)(5,10), (1,3,5,7,9)(2,4,6,8,10)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$120$:  $S_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 5: $S_5$

Low degree siblings

10T22, 12T123 x 2, 20T62 x 2, 20T65 x 2, 20T70, 24T570, 24T577, 30T58 x 2, 30T60 x 2, 40T173 x 2, 40T180, 40T181, 40T187 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

LabelCycle TypeSizeOrderRepresentative
$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $1$ $1$ $()$
$ 2, 2, 1, 1, 1, 1, 1, 1 $ $10$ $2$ $( 4,10)( 5, 9)$
$ 3, 3, 1, 1, 1, 1 $ $20$ $3$ $( 3, 5, 9)( 4, 8,10)$
$ 2, 2, 2, 2, 1, 1 $ $15$ $2$ $( 2, 4)( 3, 5)( 7, 9)( 8,10)$
$ 4, 4, 1, 1 $ $30$ $4$ $( 2, 4, 8,10)( 3, 5, 7, 9)$
$ 2, 2, 2, 2, 2 $ $15$ $2$ $( 1, 2)( 3, 4)( 5,10)( 6, 7)( 8, 9)$
$ 6, 2, 2 $ $20$ $6$ $( 1, 2)( 3, 4, 5, 8, 9,10)( 6, 7)$
$ 2, 2, 2, 2, 2 $ $10$ $2$ $( 1, 2)( 3, 8)( 4, 9)( 5,10)( 6, 7)$
$ 4, 4, 2 $ $30$ $4$ $( 1, 2, 3, 4)( 5,10)( 6, 7, 8, 9)$
$ 10 $ $24$ $10$ $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10)$
$ 6, 2, 2 $ $20$ $6$ $( 1, 2, 3, 6, 7, 8)( 4, 9)( 5,10)$
$ 3, 3, 2, 2 $ $20$ $6$ $( 1, 3)( 2, 4,10)( 5, 7, 9)( 6, 8)$
$ 5, 5 $ $24$ $5$ $( 1, 3, 5, 7, 9)( 2, 4, 6, 8,10)$
$ 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 6)( 2, 7)( 3, 8)( 4, 9)( 5,10)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $240=2^{4} \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  240.189
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 2D 2E 3A 4A 4B 5A 6A 6B 6C 10A
Size 1 1 10 10 15 15 20 30 30 24 20 20 20 24
2 P 1A 1A 1A 1A 1A 1A 3A 2D 2D 5A 3A 3A 3A 5A
3 P 1A 2A 2B 2C 2D 2E 1A 4A 4B 5A 2A 2B 2C 10A
5 P 1A 2A 2B 2C 2D 2E 3A 4A 4B 1A 6A 6B 6C 2A
Type
240.189.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1
240.189.1b R 1 −1 −1 1 1 −1 1 1 −1 1 −1 −1 1 −1
240.189.1c R 1 −1 1 −1 1 −1 1 −1 1 1 −1 1 −1 −1
240.189.1d R 1 1 −1 −1 1 1 1 −1 −1 1 1 −1 −1 1
240.189.4a R 4 4 2 2 0 0 1 0 0 −1 1 −1 −1 −1
240.189.4b R 4 4 −2 −2 0 0 1 0 0 −1 1 1 1 −1
240.189.4c R 4 −4 −2 2 0 0 1 0 0 −1 −1 1 −1 1
240.189.4d R 4 −4 2 −2 0 0 1 0 0 −1 −1 −1 1 1
240.189.5a R 5 5 −1 −1 1 1 −1 1 1 0 −1 −1 −1 0
240.189.5b R 5 5 1 1 1 1 −1 −1 −1 0 −1 1 1 0
240.189.5c R 5 −5 −1 1 1 −1 −1 −1 1 0 1 −1 1 0
240.189.5d R 5 −5 1 −1 1 −1 −1 1 −1 0 1 1 −1 0
240.189.6a R 6 6 0 0 −2 −2 0 0 0 1 0 0 0 1
240.189.6b R 6 −6 0 0 −2 2 0 0 0 1 0 0 0 −1

magma: CharacterTable(G);