Show commands:
Magma
magma: G := TransitiveGroup(33, 9);
Group action invariants
Degree $n$: | $33$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
Transitive number $t$: | $9$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
Group: | $C_{33}:C_{10}$ | ||
Parity: | $1$ | magma: IsEven(G);
| |
Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
$\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
Generators: | (1,27,21,6,18,15,24,29,10,33)(2,26,19,5,16,14,22,28,11,32)(3,25,20,4,17,13,23,30,12,31)(8,9), (1,32)(2,31)(3,33)(4,28)(5,30)(6,29)(7,25)(8,27)(9,26)(10,22)(11,24)(12,23)(13,19)(14,21)(15,20)(16,18) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $5$: $C_5$ $6$: $S_3$ $10$: $C_{10}$ $30$: $S_3 \times C_5$ $110$: $F_{11}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 11: $F_{11}$
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, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $1$ | $1$ | $()$ |
$ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ | $11$ | $5$ | $( 4,10,30,18,13)( 5,11,28,16,14)( 6,12,29,17,15)( 7,19,22,32,26) ( 8,20,23,33,27)( 9,21,24,31,25)$ |
$ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ | $11$ | $5$ | $( 4,13,18,30,10)( 5,14,16,28,11)( 6,15,17,29,12)( 7,26,32,22,19) ( 8,27,33,23,20)( 9,25,31,24,21)$ |
$ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ | $11$ | $5$ | $( 4,18,10,13,30)( 5,16,11,14,28)( 6,17,12,15,29)( 7,32,19,26,22) ( 8,33,20,27,23)( 9,31,21,25,24)$ |
$ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ | $11$ | $5$ | $( 4,30,13,10,18)( 5,28,14,11,16)( 6,29,15,12,17)( 7,22,26,19,32) ( 8,23,27,20,33)( 9,24,25,21,31)$ |
$ 10, 10, 10, 2, 1 $ | $33$ | $10$ | $( 2, 3)( 4, 9,13,25,18,31,30,24,10,21)( 5, 8,14,27,16,33,28,23,11,20) ( 6, 7,15,26,17,32,29,22,12,19)$ |
$ 10, 10, 10, 2, 1 $ | $33$ | $10$ | $( 2, 3)( 4,21,10,24,30,31,18,25,13, 9)( 5,20,11,23,28,33,16,27,14, 8) ( 6,19,12,22,29,32,17,26,15, 7)$ |
$ 10, 10, 10, 2, 1 $ | $33$ | $10$ | $( 2, 3)( 4,24,18, 9,10,31,13,21,30,25)( 5,23,16, 8,11,33,14,20,28,27) ( 6,22,17, 7,12,32,15,19,29,26)$ |
$ 10, 10, 10, 2, 1 $ | $33$ | $10$ | $( 2, 3)( 4,25,30,21,13,31,10, 9,18,24)( 5,27,28,20,14,33,11, 8,16,23) ( 6,26,29,19,15,32,12, 7,17,22)$ |
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $33$ | $2$ | $( 2, 3)( 4,31)( 5,33)( 6,32)( 7,29)( 8,28)( 9,30)(10,25)(11,27)(12,26)(13,24) (14,23)(15,22)(16,20)(17,19)(18,21)$ |
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)(31,32,33)$ |
$ 15, 15, 3 $ | $22$ | $15$ | $( 1, 2, 3)( 4,11,29,18,14, 6,10,28,17,13, 5,12,30,16,15)( 7,20,24,32,27, 9,19, 23,31,26, 8,21,22,33,25)$ |
$ 15, 15, 3 $ | $22$ | $15$ | $( 1, 2, 3)( 4,14,17,30,11, 6,13,16,29,10, 5,15,18,28,12)( 7,27,31,22,20, 9,26, 33,24,19, 8,25,32,23,21)$ |
$ 15, 15, 3 $ | $22$ | $15$ | $( 1, 2, 3)( 4,16,12,13,28, 6,18,11,15,30, 5,17,10,14,29)( 7,33,21,26,23, 9,32, 20,25,22, 8,31,19,27,24)$ |
$ 15, 15, 3 $ | $22$ | $15$ | $( 1, 2, 3)( 4,28,15,10,16, 6,30,14,12,18, 5,29,13,11,17)( 7,23,25,19,33, 9,22, 27,21,32, 8,24,26,20,31)$ |
$ 11, 11, 11 $ | $10$ | $11$ | $( 1, 4, 9,10,13,18,21,24,25,30,31)( 2, 5, 7,11,14,16,19,22,26,28,32) ( 3, 6, 8,12,15,17,20,23,27,29,33)$ |
$ 33 $ | $10$ | $33$ | $( 1, 5, 8,10,14,17,21,22,27,30,32, 3, 4, 7,12,13,16,20,24,26,29,31, 2, 6, 9, 11,15,18,19,23,25,28,33)$ |
$ 33 $ | $10$ | $33$ | $( 1, 6, 7,10,15,16,21,23,26,30,33, 2, 4, 8,11,13,17,19,24,27,28,31, 3, 5, 9, 12,14,18,20,22,25,29,32)$ |
magma: ConjugacyClasses(G);
Group invariants
Order: | $330=2 \cdot 3 \cdot 5 \cdot 11$ | magma: Order(G);
| |
Cyclic: | no | magma: IsCyclic(G);
| |
Abelian: | no | magma: IsAbelian(G);
| |
Solvable: | yes | magma: IsSolvable(G);
| |
Nilpotency class: | not nilpotent | ||
Label: | 330.3 | magma: IdentifyGroup(G);
|
Character table: |
2 1 1 1 1 1 1 1 1 1 1 . . . . . . . . 3 1 1 1 1 1 . . . . . 1 1 1 1 1 1 1 1 5 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 . . . 11 1 . . . . . . . . . 1 . . . . 1 1 1 1a 5a 5b 5c 5d 10a 10b 10c 10d 2a 3a 15a 15b 15c 15d 11a 33a 33b 2P 1a 5d 5c 5a 5b 5b 5a 5c 5d 1a 3a 15d 15c 15a 15b 11a 33a 33b 3P 1a 5c 5d 5b 5a 10d 10c 10a 10b 2a 1a 5c 5d 5b 5a 11a 11a 11a 5P 1a 1a 1a 1a 1a 2a 2a 2a 2a 2a 3a 3a 3a 3a 3a 11a 33b 33a 7P 1a 5d 5c 5a 5b 10c 10d 10b 10a 2a 3a 15d 15c 15a 15b 11a 33b 33a 11P 1a 5a 5b 5c 5d 10a 10b 10c 10d 2a 3a 15a 15b 15c 15d 1a 3a 3a 13P 1a 5c 5d 5b 5a 10d 10c 10a 10b 2a 3a 15c 15d 15b 15a 11a 33b 33a 17P 1a 5d 5c 5a 5b 10c 10d 10b 10a 2a 3a 15d 15c 15a 15b 11a 33a 33b 19P 1a 5b 5a 5d 5c 10b 10a 10d 10c 2a 3a 15b 15a 15d 15c 11a 33b 33a 23P 1a 5c 5d 5b 5a 10d 10c 10a 10b 2a 3a 15c 15d 15b 15a 11a 33b 33a 29P 1a 5b 5a 5d 5c 10b 10a 10d 10c 2a 3a 15b 15a 15d 15c 11a 33a 33b 31P 1a 5a 5b 5c 5d 10a 10b 10c 10d 2a 3a 15a 15b 15c 15d 11a 33a 33b X.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 X.3 1 A /A /B B -B -/B -/A -A -1 1 A /A /B B 1 1 1 X.4 1 B /B A /A -/A -A -/B -B -1 1 B /B A /A 1 1 1 X.5 1 /B B /A A -A -/A -B -/B -1 1 /B B /A A 1 1 1 X.6 1 /A A B /B -/B -B -A -/A -1 1 /A A B /B 1 1 1 X.7 1 A /A /B B B /B /A A 1 1 A /A /B B 1 1 1 X.8 1 B /B A /A /A A /B B 1 1 B /B A /A 1 1 1 X.9 1 /B B /A A A /A B /B 1 1 /B B /A A 1 1 1 X.10 1 /A A B /B /B B A /A 1 1 /A A B /B 1 1 1 X.11 2 2 2 2 2 . . . . . -1 -1 -1 -1 -1 2 -1 -1 X.12 2 C /C /D D . . . . . -1 -B -/B -A -/A 2 -1 -1 X.13 2 /C C D /D . . . . . -1 -/B -B -/A -A 2 -1 -1 X.14 2 D /D C /C . . . . . -1 -/A -A -B -/B 2 -1 -1 X.15 2 /D D /C C . . . . . -1 -A -/A -/B -B 2 -1 -1 X.16 10 . . . . . . . . . 10 . . . . -1 -1 -1 X.17 10 . . . . . . . . . -5 . . . . -1 E *E X.18 10 . . . . . . . . . -5 . . . . -1 *E E A = E(5)^4 B = E(5)^3 C = 2*E(5)^3 D = 2*E(5) E = E(33)^5+E(33)^7+E(33)^10+E(33)^13+E(33)^14+E(33)^19+E(33)^20+E(33)^23+E(33)^26+E(33)^28 = (1-Sqrt(33))/2 = -b33 |
magma: CharacterTable(G);