Properties

Label 33T3
Order \(66\)
n \(33\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{33}$

Learn more about

Group action invariants

Degree $n$ :  $33$
Transitive number $t$ :  $3$
Group :  $D_{33}$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
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)
$|\Aut(F/K)|$:  $1$

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 TypeSizeOrderRepresentative
$ 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 $ $2$ $33$ $( 1,18,33,13,28,10,26, 7,23, 4,20, 3,17,32,15,30,12,25, 9,22, 6,19, 2,16,31, 14,29,11,27, 8,24, 5,21)$

Group invariants

Order:  $66=2 \cdot 3 \cdot 11$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [66, 3]
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