Properties

Label 33T9
Order \(330\)
n \(33\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{33}:C_5$

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Group action invariants

Degree $n$ :  $33$
Transitive number $t$ :  $9$
Group :  $D_{33}:C_5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
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)
$|\Aut(F/K)|$:  $1$

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

Group invariants

Order:  $330=2 \cdot 3 \cdot 5 \cdot 11$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [330, 3]
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