Show commands:
Magma
magma: G := TransitiveGroup(22, 5);
Group action invariants
Degree $n$: | $22$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
Transitive number $t$: | $5$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
Group: | $C_{11}:C_{10}$ | ||
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: | (1,16,7,22,13,5,19,12,3,18,10,2,15,8,21,14,6,20,11,4,17,9), (1,11,17,3,21)(2,12,18,4,22)(5,9,8,20,14)(6,10,7,19,13) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $5$: $C_5$ $10$: $C_{10}$ $55$: $C_{11}:C_5$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 11: $C_{11}:C_5$
Low degree siblings
There are no siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 5, 5, 5, 5, 1, 1 $ | $11$ | $5$ | $( 3, 7,19,11,10)( 4, 8,20,12, 9)( 5,14,16,22,18)( 6,13,15,21,17)$ |
$ 5, 5, 5, 5, 1, 1 $ | $11$ | $5$ | $( 3,10,11,19, 7)( 4, 9,12,20, 8)( 5,18,22,16,14)( 6,17,21,15,13)$ |
$ 5, 5, 5, 5, 1, 1 $ | $11$ | $5$ | $( 3,11, 7,10,19)( 4,12, 8, 9,20)( 5,22,14,18,16)( 6,21,13,17,15)$ |
$ 5, 5, 5, 5, 1, 1 $ | $11$ | $5$ | $( 3,19,10, 7,11)( 4,20, 9, 8,12)( 5,16,18,14,22)( 6,15,17,13,21)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)$ |
$ 10, 10, 2 $ | $11$ | $10$ | $( 1, 2)( 3, 8,19,12,10, 4, 7,20,11, 9)( 5,13,16,21,18, 6,14,15,22,17)$ |
$ 10, 10, 2 $ | $11$ | $10$ | $( 1, 2)( 3, 9,11,20, 7, 4,10,12,19, 8)( 5,17,22,15,14, 6,18,21,16,13)$ |
$ 10, 10, 2 $ | $11$ | $10$ | $( 1, 2)( 3,12, 7, 9,19, 4,11, 8,10,20)( 5,21,14,17,16, 6,22,13,18,15)$ |
$ 10, 10, 2 $ | $11$ | $10$ | $( 1, 2)( 3,20,10, 8,11, 4,19, 9, 7,12)( 5,15,18,13,22, 6,16,17,14,21)$ |
$ 11, 11 $ | $5$ | $11$ | $( 1, 3, 6, 7,10,11,13,15,17,19,21)( 2, 4, 5, 8, 9,12,14,16,18,20,22)$ |
$ 22 $ | $5$ | $22$ | $( 1, 4, 6, 8,10,12,13,16,17,20,21, 2, 3, 5, 7, 9,11,14,15,18,19,22)$ |
$ 22 $ | $5$ | $22$ | $( 1, 5,10,14,17,22, 3, 8,11,16,19, 2, 6, 9,13,18,21, 4, 7,12,15,20)$ |
$ 11, 11 $ | $5$ | $11$ | $( 1, 6,10,13,17,21, 3, 7,11,15,19)( 2, 5, 9,14,18,22, 4, 8,12,16,20)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $110=2 \cdot 5 \cdot 11$ | magma: Order(G);
| |
Cyclic: | no | magma: IsCyclic(G);
| |
Abelian: | no | magma: IsAbelian(G);
| |
Solvable: | yes | magma: IsSolvable(G);
| |
Nilpotency class: | not nilpotent | ||
Label: | 110.2 | magma: IdentifyGroup(G);
|
Character table: |
2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5 1 1 1 1 1 1 1 1 1 1 . . . . 11 1 . . . . 1 . . . . 1 1 1 1 1a 5a 5b 5c 5d 2a 10a 10b 10c 10d 11a 22a 22b 11b 2P 1a 5d 5c 5a 5b 1a 5d 5c 5a 5b 11b 11b 11a 11a 3P 1a 5c 5d 5b 5a 2a 10c 10d 10b 10a 11a 22a 22b 11b 5P 1a 1a 1a 1a 1a 2a 2a 2a 2a 2a 11a 22a 22b 11b 7P 1a 5d 5c 5a 5b 2a 10d 10c 10a 10b 11b 22b 22a 11a 11P 1a 5a 5b 5c 5d 2a 10a 10b 10c 10d 1a 2a 2a 1a 13P 1a 5c 5d 5b 5a 2a 10c 10d 10b 10a 11b 22b 22a 11a 17P 1a 5d 5c 5a 5b 2a 10d 10c 10a 10b 11b 22b 22a 11a 19P 1a 5b 5a 5d 5c 2a 10b 10a 10d 10c 11b 22b 22a 11a 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 A /A /B B -1 -A -/A -/B -B 1 -1 -1 1 X.4 1 B /B A /A -1 -B -/B -A -/A 1 -1 -1 1 X.5 1 /B B /A A -1 -/B -B -/A -A 1 -1 -1 1 X.6 1 /A A B /B -1 -/A -A -B -/B 1 -1 -1 1 X.7 1 A /A /B B 1 A /A /B B 1 1 1 1 X.8 1 B /B A /A 1 B /B A /A 1 1 1 1 X.9 1 /B B /A A 1 /B B /A A 1 1 1 1 X.10 1 /A A B /B 1 /A A B /B 1 1 1 1 X.11 5 . . . . -5 . . . . C -C -/C /C X.12 5 . . . . -5 . . . . /C -/C -C C X.13 5 . . . . 5 . . . . C C /C /C X.14 5 . . . . 5 . . . . /C /C C C A = E(5)^4 B = E(5)^3 C = E(11)^2+E(11)^6+E(11)^7+E(11)^8+E(11)^10 = (-1-Sqrt(-11))/2 = -1-b11 |
magma: CharacterTable(G);