Show commands:
Magma
magma: G := TransitiveGroup(22, 27);
Group action invariants
Degree $n$: | $22$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
Transitive number $t$: | $27$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
Group: | $C_2\times M_{11}$ | ||
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: | (1,22,17,16,12,5,3,7,14,9,19,2,21,18,15,11,6,4,8,13,10,20), (1,4,10,5,19,22)(2,3,9,6,20,21)(7,17,13,8,18,14)(11,15)(12,16) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $7920$: $M_{11}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 11: $M_{11}$
Low degree siblings
22T26, 24T12204, 44T140Siblings 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $1$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)$ |
$ 5, 5, 5, 5, 1, 1 $ | $1584$ | $5$ | $( 3,14, 8, 6,10)( 4,13, 7, 5, 9)(11,16,20,22,18)(12,15,19,21,17)$ |
$ 10, 10, 2 $ | $1584$ | $10$ | $( 1, 2)( 3,13, 8, 5,10, 4,14, 7, 6, 9)(11,15,20,21,18,12,16,19,22,17)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $ | $165$ | $2$ | $( 1, 6)( 2, 5)( 3,17)( 4,18)( 9,11)(10,12)(13,20)(14,19)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $165$ | $2$ | $( 1, 5)( 2, 6)( 3,18)( 4,17)( 7, 8)( 9,12)(10,11)(13,19)(14,20)(15,16)(21,22)$ |
$ 3, 3, 3, 3, 3, 3, 1, 1, 1, 1 $ | $440$ | $3$ | $( 3,14,12)( 4,13,11)( 7,22,16)( 8,21,15)( 9,18,20)(10,17,19)$ |
$ 6, 6, 6, 2, 2 $ | $440$ | $6$ | $( 1, 2)( 3,13,12, 4,14,11)( 5, 6)( 7,21,16, 8,22,15)( 9,17,20,10,18,19)$ |
$ 6, 6, 3, 3, 2, 2 $ | $1320$ | $6$ | $( 1, 6)( 2, 5)( 3,10,14,17,12,19)( 4, 9,13,18,11,20)( 7,16,22)( 8,15,21)$ |
$ 6, 6, 6, 2, 2 $ | $1320$ | $6$ | $( 1, 5)( 2, 6)( 3, 9,14,18,12,20)( 4,10,13,17,11,19)( 7,15,22, 8,16,21)$ |
$ 4, 4, 4, 4, 1, 1, 1, 1, 1, 1 $ | $990$ | $4$ | $( 3,12,10,21)( 4,11, 9,22)( 5,20,13,16)( 6,19,14,15)$ |
$ 4, 4, 4, 4, 2, 2, 2 $ | $990$ | $4$ | $( 1, 2)( 3,11,10,22)( 4,12, 9,21)( 5,19,13,15)( 6,20,14,16)( 7, 8)(17,18)$ |
$ 8, 8, 2, 2, 2 $ | $990$ | $8$ | $( 1, 2)( 3,20,12,13,10,16,21, 5)( 4,19,11,14, 9,15,22, 6)( 7,17)( 8,18)$ |
$ 8, 8, 2, 2, 1, 1 $ | $990$ | $8$ | $( 3,19,12,14,10,15,21, 6)( 4,20,11,13, 9,16,22, 5)( 7,18)( 8,17)$ |
$ 8, 8, 2, 2, 2 $ | $990$ | $8$ | $( 1, 2)( 3, 5,21,16,10,13,12,20)( 4, 6,22,15, 9,14,11,19)( 7,17)( 8,18)$ |
$ 8, 8, 2, 2, 1, 1 $ | $990$ | $8$ | $( 3, 6,21,15,10,14,12,19)( 4, 5,22,16, 9,13,11,20)( 7,18)( 8,17)$ |
$ 11, 11 $ | $720$ | $11$ | $( 1,15, 6, 8,14,17,12,19,10,21, 3)( 2,16, 5, 7,13,18,11,20, 9,22, 4)$ |
$ 22 $ | $720$ | $22$ | $( 1,16, 6, 7,14,18,12,20,10,22, 3, 2,15, 5, 8,13,17,11,19, 9,21, 4)$ |
$ 22 $ | $720$ | $22$ | $( 1, 4,21, 9,19,11,17,13, 8, 5,15, 2, 3,22,10,20,12,18,14, 7, 6,16)$ |
$ 11, 11 $ | $720$ | $11$ | $( 1, 3,21,10,19,12,17,14, 8, 6,15)( 2, 4,22, 9,20,11,18,13, 7, 5,16)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $15840=2^{5} \cdot 3^{2} \cdot 5 \cdot 11$ | magma: Order(G);
| |
Cyclic: | no | magma: IsCyclic(G);
| |
Abelian: | no | magma: IsAbelian(G);
| |
Solvable: | no | magma: IsSolvable(G);
| |
Nilpotency class: | not nilpotent | ||
Label: | 15840.q | magma: IdentifyGroup(G);
|
Character table: |
2 5 5 1 1 5 5 4 4 2 2 2 2 4 4 4 4 1 1 1 1 3 2 2 . . 1 1 . . 2 2 1 1 . . . . . . . . 5 1 1 1 1 . . . . . . . . . . . . . . . . 11 1 1 . . . . . . . . . . . . . . 1 1 1 1 1a 2a 5a 10a 2b 2c 4a 4b 6a 3a 6b 6c 8a 8b 8c 8d 22a 11a 11b 22b 2P 1a 1a 5a 5a 1a 1a 2b 2b 3a 3a 3a 3a 4b 4b 4b 4b 11b 11b 11a 11a 3P 1a 2a 5a 10a 2b 2c 4a 4b 2a 1a 2c 2b 8a 8b 8c 8d 22a 11a 11b 22b 5P 1a 2a 1a 2a 2b 2c 4a 4b 6a 3a 6b 6c 8d 8c 8b 8a 22a 11a 11b 22b 7P 1a 2a 5a 10a 2b 2c 4a 4b 6a 3a 6b 6c 8d 8c 8b 8a 22b 11b 11a 22a 11P 1a 2a 5a 10a 2b 2c 4a 4b 6a 3a 6b 6c 8a 8b 8c 8d 2a 1a 1a 2a 13P 1a 2a 5a 10a 2b 2c 4a 4b 6a 3a 6b 6c 8d 8c 8b 8a 22b 11b 11a 22a 17P 1a 2a 5a 10a 2b 2c 4a 4b 6a 3a 6b 6c 8a 8b 8c 8d 22b 11b 11a 22a 19P 1a 2a 5a 10a 2b 2c 4a 4b 6a 3a 6b 6c 8a 8b 8c 8d 22b 11b 11a 22a X.1 1 1 1 1 1 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 -1 1 -1 1 1 -1 X.3 10 10 . . 2 2 2 2 1 1 -1 -1 . . . . -1 -1 -1 -1 X.4 10 -10 . . 2 -2 -2 2 -1 1 1 -1 . . . . 1 -1 -1 1 X.5 10 -10 . . -2 2 . . -1 1 -1 1 A -A A -A 1 -1 -1 1 X.6 10 -10 . . -2 2 . . -1 1 -1 1 -A A -A A 1 -1 -1 1 X.7 10 10 . . -2 -2 . . 1 1 1 1 A A -A -A -1 -1 -1 -1 X.8 10 10 . . -2 -2 . . 1 1 1 1 -A -A A A -1 -1 -1 -1 X.9 11 11 1 1 3 3 -1 -1 2 2 . . -1 -1 -1 -1 . . . . X.10 11 -11 1 -1 3 -3 1 -1 -2 2 . . -1 1 1 -1 . . . . X.11 16 -16 1 -1 . . . . 2 -2 . . . . . . B -B -/B /B X.12 16 -16 1 -1 . . . . 2 -2 . . . . . . /B -/B -B B X.13 16 16 1 1 . . . . -2 -2 . . . . . . -/B -/B -B -B X.14 16 16 1 1 . . . . -2 -2 . . . . . . -B -B -/B -/B X.15 44 44 -1 -1 4 4 . . -1 -1 1 1 . . . . . . . . X.16 44 -44 -1 1 4 -4 . . 1 -1 -1 1 . . . . . . . . X.17 45 45 . . -3 -3 1 1 . . . . -1 -1 -1 -1 1 1 1 1 X.18 45 -45 . . -3 3 -1 1 . . . . -1 1 1 -1 -1 1 1 -1 X.19 55 55 . . -1 -1 -1 -1 1 1 -1 -1 1 1 1 1 . . . . X.20 55 -55 . . -1 1 1 -1 -1 1 1 -1 1 -1 -1 1 . . . . A = -E(8)-E(8)^3 = -Sqrt(-2) = -i2 B = -E(11)-E(11)^3-E(11)^4-E(11)^5-E(11)^9 = (1-Sqrt(-11))/2 = -b11 |
magma: CharacterTable(G);