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 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
Cycle Type | Size | Order | Representative |
$ 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: |
2 4 3 2 4 3 4 2 3 3 1 2 2 1 4 3 1 1 1 . . . 1 1 . . 1 1 . 1 5 1 . . . . . . . . 1 . . 1 1 1a 2a 3a 2b 4a 2c 6a 2d 4b 10a 6b 6c 5a 2e 2P 1a 1a 3a 1a 2b 1a 3a 1a 2b 5a 3a 3a 5a 1a 3P 1a 2a 1a 2b 4a 2c 2d 2d 4b 10a 2e 2a 5a 2e 5P 1a 2a 3a 2b 4a 2c 6a 2d 4b 2e 6b 6c 1a 2e 7P 1a 2a 3a 2b 4a 2c 6a 2d 4b 10a 6b 6c 5a 2e X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 1 -1 -1 1 1 1 -1 -1 -1 1 -1 X.3 1 -1 1 1 -1 1 -1 -1 -1 1 1 -1 1 1 X.4 1 1 1 1 1 -1 -1 -1 -1 -1 -1 1 1 -1 X.5 4 -2 1 . . . 1 -2 . -1 1 1 -1 4 X.6 4 2 1 . . . -1 2 . -1 1 -1 -1 4 X.7 4 -2 1 . . . -1 2 . 1 -1 1 -1 -4 X.8 4 2 1 . . . 1 -2 . 1 -1 -1 -1 -4 X.9 5 1 -1 1 -1 1 1 1 -1 . -1 1 . 5 X.10 5 -1 -1 1 1 1 -1 -1 1 . -1 -1 . 5 X.11 5 1 -1 1 -1 -1 -1 -1 1 . 1 1 . -5 X.12 5 -1 -1 1 1 -1 1 1 -1 . 1 -1 . -5 X.13 6 . . -2 . -2 . . . 1 . . 1 6 X.14 6 . . -2 . 2 . . . -1 . . 1 -6 |
magma: CharacterTable(G);