LMFDB - Galois group: 33T3

Show commands: Magma

magma: G := TransitiveGroup(33, 3);

Group action invariants

Degree $n$:	$33$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$3$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$D_{33}$
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:	(1,10,20,30,6,14,24,33,7,17,25,2,11,21,28,4,15,22,31,8,18,26,3,12,19,29,5,13,23,32,9,16,27), (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);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$
$6$: $S_3$
$22$: $D_{11}$

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$
$6$:	$S_3$
$22$:	$D_{11}$

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

Label	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);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	not nilpotent
Label:	66.3	magma: IdentifyGroup(G);
Character table:

Size

11A5

11A1

11A4

11A2

11A3

33A2

33A5

33A13

33A4

33A16

33A10

33A8

33A1

33A14

33A7

3 P

11A2

11A4

11A5

11A3

11A1

11A3

11A1

11A2

11A3

11A5

11A4

11A5

11A4

11A2

11 P

11A4

11A3

11A1

11A5

11A2

33A5

33A4

33A16

33A10

33A7

33A8

33A13

33A14

33A2

33A1

Type

66.3.1a

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

66.3.1b

1

- 1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

1

66.3.2a

2

0

- 1

2

2

2

2

2

- 1

- 1

- 1

- 1

- 1

- 1

- 1

- 1

- 1

- 1

66.3.2b1

2

0

2

ζ_{11}^{- 5} + ζ_{11}^{5}

ζ_{11}^{- 1} + ζ_{11}

ζ_{11}^{- 4} + ζ_{11}^{4}

ζ_{11}^{- 2} + ζ_{11}^{2}

ζ_{11}^{- 3} + ζ_{11}^{3}

ζ_{11}^{- 2} + ζ_{11}^{2}

ζ_{11}^{- 4} + ζ_{11}^{4}

ζ_{11}^{- 3} + ζ_{11}^{3}

ζ_{11}^{- 1} + ζ_{11}

ζ_{11}^{- 3} + ζ_{11}^{3}

ζ_{11}^{- 5} + ζ_{11}^{5}

ζ_{11}^{- 2} + ζ_{11}^{2}

ζ_{11}^{- 4} + ζ_{11}^{4}

ζ_{11}^{- 5} + ζ_{11}^{5}

ζ_{11}^{- 1} + ζ_{11}

66.3.2b2

2

0

2

ζ_{11}^{- 4} + ζ_{11}^{4}

ζ_{11}^{- 3} + ζ_{11}^{3}

ζ_{11}^{- 1} + ζ_{11}

ζ_{11}^{- 5} + ζ_{11}^{5}

ζ_{11}^{- 2} + ζ_{11}^{2}

ζ_{11}^{- 5} + ζ_{11}^{5}

ζ_{11}^{- 1} + ζ_{11}

ζ_{11}^{- 2} + ζ_{11}^{2}

ζ_{11}^{- 3} + ζ_{11}^{3}

ζ_{11}^{- 2} + ζ_{11}^{2}

ζ_{11}^{- 4} + ζ_{11}^{4}

ζ_{11}^{- 5} + ζ_{11}^{5}

ζ_{11}^{- 1} + ζ_{11}

ζ_{11}^{- 4} + ζ_{11}^{4}

ζ_{11}^{- 3} + ζ_{11}^{3}

66.3.2b3

2

0

2

ζ_{11}^{- 3} + ζ_{11}^{3}

ζ_{11}^{- 5} + ζ_{11}^{5}

ζ_{11}^{- 2} + ζ_{11}^{2}

ζ_{11}^{- 1} + ζ_{11}

ζ_{11}^{- 4} + ζ_{11}^{4}

ζ_{11}^{- 1} + ζ_{11}

ζ_{11}^{- 2} + ζ_{11}^{2}

ζ_{11}^{- 4} + ζ_{11}^{4}

ζ_{11}^{- 5} + ζ_{11}^{5}

ζ_{11}^{- 4} + ζ_{11}^{4}

ζ_{11}^{- 3} + ζ_{11}^{3}

ζ_{11}^{- 1} + ζ_{11}

ζ_{11}^{- 2} + ζ_{11}^{2}

ζ_{11}^{- 3} + ζ_{11}^{3}

ζ_{11}^{- 5} + ζ_{11}^{5}

66.3.2b4

2

0

2

ζ_{11}^{- 2} + ζ_{11}^{2}

ζ_{11}^{- 4} + ζ_{11}^{4}

ζ_{11}^{- 5} + ζ_{11}^{5}

ζ_{11}^{- 3} + ζ_{11}^{3}

ζ_{11}^{- 1} + ζ_{11}

ζ_{11}^{- 3} + ζ_{11}^{3}

ζ_{11}^{- 5} + ζ_{11}^{5}

ζ_{11}^{- 1} + ζ_{11}

ζ_{11}^{- 4} + ζ_{11}^{4}

ζ_{11}^{- 1} + ζ_{11}

ζ_{11}^{- 2} + ζ_{11}^{2}

ζ_{11}^{- 3} + ζ_{11}^{3}

ζ_{11}^{- 5} + ζ_{11}^{5}

ζ_{11}^{- 2} + ζ_{11}^{2}

ζ_{11}^{- 4} + ζ_{11}^{4}

66.3.2b5

2

0

2

ζ_{11}^{- 1} + ζ_{11}

ζ_{11}^{- 2} + ζ_{11}^{2}

ζ_{11}^{- 3} + ζ_{11}^{3}

ζ_{11}^{- 4} + ζ_{11}^{4}

ζ_{11}^{- 5} + ζ_{11}^{5}

ζ_{11}^{- 4} + ζ_{11}^{4}

ζ_{11}^{- 3} + ζ_{11}^{3}

ζ_{11}^{- 5} + ζ_{11}^{5}

ζ_{11}^{- 2} + ζ_{11}^{2}

ζ_{11}^{- 5} + ζ_{11}^{5}

ζ_{11}^{- 1} + ζ_{11}

ζ_{11}^{- 4} + ζ_{11}^{4}

ζ_{11}^{- 3} + ζ_{11}^{3}

ζ_{11}^{- 1} + ζ_{11}

ζ_{11}^{- 2} + ζ_{11}^{2}

66.3.2c1

2

0

- 1

ζ_{33}^{- 15} + ζ_{33}^{15}

ζ_{33}^{- 3} + ζ_{33}^{3}

ζ_{33}^{- 12} + ζ_{33}^{12}

ζ_{33}^{- 6} + ζ_{33}^{6}

ζ_{33}^{- 9} + ζ_{33}^{9}

ζ_{33}^{- 16} + ζ_{33}^{16}

ζ_{33}^{- 1} + ζ_{33}

ζ_{33}^{- 2} + ζ_{33}^{2}

ζ_{33}^{- 14} + ζ_{33}^{14}

ζ_{33}^{- 13} + ζ_{33}^{13}

ζ_{33}^{- 4} + ζ_{33}^{4}

ζ_{33}^{- 5} + ζ_{33}^{5}

ζ_{33}^{- 10} + ζ_{33}^{10}

ζ_{33}^{- 7} + ζ_{33}^{7}

ζ_{33}^{- 8} + ζ_{33}^{8}

66.3.2c2

2

0

- 1

ζ_{33}^{- 15} + ζ_{33}^{15}

ζ_{33}^{- 3} + ζ_{33}^{3}

ζ_{33}^{- 12} + ζ_{33}^{12}

ζ_{33}^{- 6} + ζ_{33}^{6}

ζ_{33}^{- 9} + ζ_{33}^{9}

ζ_{33}^{- 5} + ζ_{33}^{5}

ζ_{33}^{- 10} + ζ_{33}^{10}

ζ_{33}^{- 13} + ζ_{33}^{13}

ζ_{33}^{- 8} + ζ_{33}^{8}

ζ_{33}^{- 2} + ζ_{33}^{2}

ζ_{33}^{- 7} + ζ_{33}^{7}

ζ_{33}^{- 16} + ζ_{33}^{16}

ζ_{33}^{- 1} + ζ_{33}

ζ_{33}^{- 4} + ζ_{33}^{4}

ζ_{33}^{- 14} + ζ_{33}^{14}

66.3.2c3

2

0

- 1

ζ_{33}^{- 12} + ζ_{33}^{12}

ζ_{33}^{- 9} + ζ_{33}^{9}

ζ_{33}^{- 3} + ζ_{33}^{3}

ζ_{33}^{- 15} + ζ_{33}^{15}

ζ_{33}^{- 6} + ζ_{33}^{6}

ζ_{33}^{- 7} + ζ_{33}^{7}

ζ_{33}^{- 14} + ζ_{33}^{14}

ζ_{33}^{- 5} + ζ_{33}^{5}

ζ_{33}^{- 2} + ζ_{33}^{2}

ζ_{33}^{- 16} + ζ_{33}^{16}

ζ_{33}^{- 10} + ζ_{33}^{10}

ζ_{33}^{- 4} + ζ_{33}^{4}

ζ_{33}^{- 8} + ζ_{33}^{8}

ζ_{33}^{- 1} + ζ_{33}

ζ_{33}^{- 13} + ζ_{33}^{13}

66.3.2c4

2

0

- 1

ζ_{33}^{- 12} + ζ_{33}^{12}

ζ_{33}^{- 9} + ζ_{33}^{9}

ζ_{33}^{- 3} + ζ_{33}^{3}

ζ_{33}^{- 15} + ζ_{33}^{15}

ζ_{33}^{- 6} + ζ_{33}^{6}

ζ_{33}^{- 4} + ζ_{33}^{4}

ζ_{33}^{- 8} + ζ_{33}^{8}

ζ_{33}^{- 16} + ζ_{33}^{16}

ζ_{33}^{- 13} + ζ_{33}^{13}

ζ_{33}^{- 5} + ζ_{33}^{5}

ζ_{33}^{- 1} + ζ_{33}

ζ_{33}^{- 7} + ζ_{33}^{7}

ζ_{33}^{- 14} + ζ_{33}^{14}

ζ_{33}^{- 10} + ζ_{33}^{10}

ζ_{33}^{- 2} + ζ_{33}^{2}

66.3.2c5

2

0

- 1

ζ_{33}^{- 9} + ζ_{33}^{9}

ζ_{33}^{- 15} + ζ_{33}^{15}

ζ_{33}^{- 6} + ζ_{33}^{6}

ζ_{33}^{- 3} + ζ_{33}^{3}

ζ_{33}^{- 12} + ζ_{33}^{12}

ζ_{33}^{- 14} + ζ_{33}^{14}

ζ_{33}^{- 5} + ζ_{33}^{5}

ζ_{33}^{- 10} + ζ_{33}^{10}

ζ_{33}^{- 4} + ζ_{33}^{4}

ζ_{33}^{- 1} + ζ_{33}

ζ_{33}^{- 13} + ζ_{33}^{13}

ζ_{33}^{- 8} + ζ_{33}^{8}

ζ_{33}^{- 16} + ζ_{33}^{16}

ζ_{33}^{- 2} + ζ_{33}^{2}

ζ_{33}^{- 7} + ζ_{33}^{7}

66.3.2c6

2

0

- 1

ζ_{33}^{- 9} + ζ_{33}^{9}

ζ_{33}^{- 15} + ζ_{33}^{15}

ζ_{33}^{- 6} + ζ_{33}^{6}

ζ_{33}^{- 3} + ζ_{33}^{3}

ζ_{33}^{- 12} + ζ_{33}^{12}

ζ_{33}^{- 8} + ζ_{33}^{8}

ζ_{33}^{- 16} + ζ_{33}^{16}

ζ_{33}^{- 1} + ζ_{33}

ζ_{33}^{- 7} + ζ_{33}^{7}

ζ_{33}^{- 10} + ζ_{33}^{10}

ζ_{33}^{- 2} + ζ_{33}^{2}

ζ_{33}^{- 14} + ζ_{33}^{14}

ζ_{33}^{- 5} + ζ_{33}^{5}

ζ_{33}^{- 13} + ζ_{33}^{13}

ζ_{33}^{- 4} + ζ_{33}^{4}

66.3.2c7

2

0

- 1

ζ_{33}^{- 6} + ζ_{33}^{6}

ζ_{33}^{- 12} + ζ_{33}^{12}

ζ_{33}^{- 15} + ζ_{33}^{15}

ζ_{33}^{- 9} + ζ_{33}^{9}

ζ_{33}^{- 3} + ζ_{33}^{3}

ζ_{33}^{- 13} + ζ_{33}^{13}

ζ_{33}^{- 7} + ζ_{33}^{7}

ζ_{33}^{- 14} + ζ_{33}^{14}

ζ_{33}^{- 1} + ζ_{33}

ζ_{33}^{- 8} + ζ_{33}^{8}

ζ_{33}^{- 5} + ζ_{33}^{5}

ζ_{33}^{- 2} + ζ_{33}^{2}

ζ_{33}^{- 4} + ζ_{33}^{4}

ζ_{33}^{- 16} + ζ_{33}^{16}

ζ_{33}^{- 10} + ζ_{33}^{10}

66.3.2c8

2

0

- 1

ζ_{33}^{- 6} + ζ_{33}^{6}

ζ_{33}^{- 12} + ζ_{33}^{12}

ζ_{33}^{- 15} + ζ_{33}^{15}

ζ_{33}^{- 9} + ζ_{33}^{9}

ζ_{33}^{- 3} + ζ_{33}^{3}

ζ_{33}^{- 2} + ζ_{33}^{2}

ζ_{33}^{- 4} + ζ_{33}^{4}

ζ_{33}^{- 8} + ζ_{33}^{8}

ζ_{33}^{- 10} + ζ_{33}^{10}

ζ_{33}^{- 14} + ζ_{33}^{14}

ζ_{33}^{- 16} + ζ_{33}^{16}

ζ_{33}^{- 13} + ζ_{33}^{13}

ζ_{33}^{- 7} + ζ_{33}^{7}

ζ_{33}^{- 5} + ζ_{33}^{5}

ζ_{33}^{- 1} + ζ_{33}

66.3.2c9

2

0

- 1

ζ_{33}^{- 3} + ζ_{33}^{3}

ζ_{33}^{- 6} + ζ_{33}^{6}

ζ_{33}^{- 9} + ζ_{33}^{9}

ζ_{33}^{- 12} + ζ_{33}^{12}

ζ_{33}^{- 15} + ζ_{33}^{15}

ζ_{33}^{- 10} + ζ_{33}^{10}

ζ_{33}^{- 13} + ζ_{33}^{13}

ζ_{33}^{- 7} + ζ_{33}^{7}

ζ_{33}^{- 16} + ζ_{33}^{16}

ζ_{33}^{- 4} + ζ_{33}^{4}

ζ_{33}^{- 14} + ζ_{33}^{14}

ζ_{33}^{- 1} + ζ_{33}

ζ_{33}^{- 2} + ζ_{33}^{2}

ζ_{33}^{- 8} + ζ_{33}^{8}

ζ_{33}^{- 5} + ζ_{33}^{5}

66.3.2c10

2

0

- 1

ζ_{33}^{- 3} + ζ_{33}^{3}

ζ_{33}^{- 6} + ζ_{33}^{6}

ζ_{33}^{- 9} + ζ_{33}^{9}

ζ_{33}^{- 12} + ζ_{33}^{12}

ζ_{33}^{- 15} + ζ_{33}^{15}

ζ_{33}^{- 1} + ζ_{33}

ζ_{33}^{- 2} + ζ_{33}^{2}

ζ_{33}^{- 4} + ζ_{33}^{4}

ζ_{33}^{- 5} + ζ_{33}^{5}

ζ_{33}^{- 7} + ζ_{33}^{7}

ζ_{33}^{- 8} + ζ_{33}^{8}

ζ_{33}^{- 10} + ζ_{33}^{10}

ζ_{33}^{- 13} + ζ_{33}^{13}

ζ_{33}^{- 14} + ζ_{33}^{14}

ζ_{33}^{- 16} + ζ_{33}^{16}

magma: CharacterTable(G);

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Group action invariants

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Group invariants

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	33T3
Degree	$33$
Order	$66$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$D_{33}$