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Magma
magma: G := TransitiveGroup(33, 3);
Group action invariants
| Degree $n$: | $33$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $3$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $D_{33}$ | ||
| Parity: | $1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,10,20,30,6,14,24,33,7,17,25,2,11,21,28,4,15,22,31,8,18,26,3,12,19,29,5,13,23,32,9,16,27), (1,32)(2,31)(3,33)(4,28)(5,30)(6,29)(7,26)(8,25)(9,27)(10,23)(11,22)(12,24)(13,20)(14,19)(15,21)(17,18) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $6$: $S_3$ $22$: $D_{11}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 3: $S_3$
Degree 11: $D_{11}$
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 $ | $1$ | $1$ | $()$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $ | $33$ | $2$ | $( 2, 3)( 4,31)( 5,33)( 6,32)( 7,29)( 8,28)( 9,30)(10,27)(11,26)(12,25)(13,24) (14,23)(15,22)(16,20)(17,19)(18,21)$ |
| $ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ | $2$ | $3$ | $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)(31,32,33)$ |
| $ 11, 11, 11 $ | $2$ | $11$ | $( 1, 4, 9,11,13,17,19,24,26,30,31)( 2, 5, 7,12,14,18,20,22,27,28,32) ( 3, 6, 8,10,15,16,21,23,25,29,33)$ |
| $ 33 $ | $2$ | $33$ | $( 1, 5, 8,11,14,16,19,22,25,30,32, 3, 4, 7,10,13,18,21,24,27,29,31, 2, 6, 9, 12,15,17,20,23,26,28,33)$ |
| $ 33 $ | $2$ | $33$ | $( 1, 6, 7,11,15,18,19,23,27,30,33, 2, 4, 8,12,13,16,20,24,25,28,31, 3, 5, 9, 10,14,17,21,22,26,29,32)$ |
| $ 33 $ | $2$ | $33$ | $( 1, 7,15,19,27,33, 4,12,16,24,28, 3, 9,14,21,26,32, 6,11,18,23,30, 2, 8,13, 20,25,31, 5,10,17,22,29)$ |
| $ 33 $ | $2$ | $33$ | $( 1, 8,14,19,25,32, 4,10,18,24,29, 2, 9,15,20,26,33, 5,11,16,22,30, 3, 7,13, 21,27,31, 6,12,17,23,28)$ |
| $ 11, 11, 11 $ | $2$ | $11$ | $( 1, 9,13,19,26,31, 4,11,17,24,30)( 2, 7,14,20,27,32, 5,12,18,22,28) ( 3, 8,15,21,25,33, 6,10,16,23,29)$ |
| $ 33 $ | $2$ | $33$ | $( 1,10,20,30, 6,14,24,33, 7,17,25, 2,11,21,28, 4,15,22,31, 8,18,26, 3,12,19, 29, 5,13,23,32, 9,16,27)$ |
| $ 11, 11, 11 $ | $2$ | $11$ | $( 1,11,19,30, 4,13,24,31, 9,17,26)( 2,12,20,28, 5,14,22,32, 7,18,27) ( 3,10,21,29, 6,15,23,33, 8,16,25)$ |
| $ 33 $ | $2$ | $33$ | $( 1,12,21,30, 5,15,24,32, 8,17,27, 3,11,20,29, 4,14,23,31, 7,16,26, 2,10,19, 28, 6,13,22,33, 9,18,25)$ |
| $ 11, 11, 11 $ | $2$ | $11$ | $( 1,13,26, 4,17,30, 9,19,31,11,24)( 2,14,27, 5,18,28, 7,20,32,12,22) ( 3,15,25, 6,16,29, 8,21,33,10,23)$ |
| $ 33 $ | $2$ | $33$ | $( 1,14,25, 4,18,29, 9,20,33,11,22, 3,13,27, 6,17,28, 8,19,32,10,24, 2,15,26, 5,16,30, 7,21,31,12,23)$ |
| $ 33 $ | $2$ | $33$ | $( 1,15,27, 4,16,28, 9,21,32,11,23, 2,13,25, 5,17,29, 7,19,33,12,24, 3,14,26, 6,18,30, 8,20,31,10,22)$ |
| $ 33 $ | $2$ | $33$ | $( 1,16,32,13,29,12,26, 8,22, 4,21, 2,17,33,14,30,10,27, 9,23, 5,19, 3,18,31, 15,28,11,25, 7,24, 6,20)$ |
| $ 11, 11, 11 $ | $2$ | $11$ | $( 1,17,31,13,30,11,26, 9,24, 4,19)( 2,18,32,14,28,12,27, 7,22, 5,20) ( 3,16,33,15,29,10,25, 8,23, 6,21)$ |
| $ 33 $ | $2$ | $33$ | $( 1,18,33,13,28,10,26, 7,23, 4,20, 3,17,32,15,30,12,25, 9,22, 6,19, 2,16,31, 14,29,11,27, 8,24, 5,21)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $66=2 \cdot 3 \cdot 11$ | magma: Order(G);
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| Cyclic: | no | magma: IsCyclic(G);
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| Abelian: | no | magma: IsAbelian(G);
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| Solvable: | yes | magma: IsSolvable(G);
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| Nilpotency class: | not nilpotent | ||
| Label: | 66.3 | magma: IdentifyGroup(G);
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| Character table: |
2 1 1 . . . . . . . . . . . . . . . .
3 1 . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
11 1 . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1a 2a 3a 11a 33a 33b 33c 33d 11b 33e 11c 33f 11d 33g 33h 33i 11e 33j
2P 1a 1a 3a 11b 33d 33c 33h 33g 11d 33i 11e 33j 11c 33f 33e 33b 11a 33a
3P 1a 2a 1a 11c 11c 11c 11e 11e 11e 11b 11b 11b 11a 11a 11a 11d 11d 11d
5P 1a 2a 3a 11e 33i 33j 33a 33b 11a 33g 11d 33h 11b 33c 33d 33f 11c 33e
7P 1a 2a 3a 11d 33h 33g 33f 33e 11c 33a 11a 33b 11e 33i 33j 33d 11b 33c
11P 1a 2a 3a 1a 3a 3a 3a 3a 1a 3a 1a 3a 1a 3a 3a 3a 1a 3a
13P 1a 2a 3a 11b 33c 33d 33g 33h 11d 33j 11e 33i 11c 33e 33f 33a 11a 33b
17P 1a 2a 3a 11e 33j 33i 33b 33a 11a 33h 11d 33g 11b 33d 33c 33e 11c 33f
19P 1a 2a 3a 11c 33e 33f 33j 33i 11e 33d 11b 33c 11a 33b 33a 33g 11d 33h
23P 1a 2a 3a 11a 33b 33a 33d 33c 11b 33f 11c 33e 11d 33h 33g 33j 11e 33i
29P 1a 2a 3a 11d 33g 33h 33e 33f 11c 33b 11a 33a 11e 33j 33i 33c 11b 33d
31P 1a 2a 3a 11b 33d 33c 33h 33g 11d 33i 11e 33j 11c 33f 33e 33b 11a 33a
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 2 . -1 2 -1 -1 -1 -1 2 -1 2 -1 2 -1 -1 -1 2 -1
X.4 2 . 2 A A A B B B C C C D D D E E E
X.5 2 . 2 B B B D D D E E E C C C A A A
X.6 2 . 2 C C C E E E B B B A A A D D D
X.7 2 . 2 D D D C C C A A A E E E B B B
X.8 2 . 2 E E E A A A D D D B B B C C C
X.9 2 . -1 C F G O N E I B H A M L K D J
X.10 2 . -1 C G F N O E H B I A L M J D K
X.11 2 . -1 B H I K J D O E N C F G L A M
X.12 2 . -1 B I H J K D N E O C G F M A L
X.13 2 . -1 D J K G F C L A M E N O I B H
X.14 2 . -1 D K J F G C M A L E O N H B I
X.15 2 . -1 A L M H I B F C G D K J N E O
X.16 2 . -1 A M L I H B G C F D J K O E N
X.17 2 . -1 E N O L M A K D J B H I G C F
X.18 2 . -1 E O N M L A J D K B I H F C G
A = E(11)^2+E(11)^9
B = E(11)^4+E(11)^7
C = E(11)^5+E(11)^6
D = E(11)^3+E(11)^8
E = E(11)+E(11)^10
F = E(33)^7+E(33)^26
G = E(33)^4+E(33)^29
H = E(33)^10+E(33)^23
I = E(33)+E(33)^32
J = E(33)^13+E(33)^20
K = E(33)^2+E(33)^31
L = E(33)^16+E(33)^17
M = E(33)^5+E(33)^28
N = E(33)^14+E(33)^19
O = E(33)^8+E(33)^25
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magma: CharacterTable(G);