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Properties

Label 33T5
Degree $33$
Order $132$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $S_3\times D_{11}$

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magma: G := TransitiveGroup(33, 5);
 

Group action invariants

Degree $n$:  $33$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $5$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\times D_{11}$
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:  (1,16,32,14,30,11,27,8,24,4,19)(2,18,33,13,28,10,25,7,22,6,20,3,17,31,15,29,12,26,9,23,5,21), (1,6,2,4,3,5)(7,33,8,31,9,32)(10,28,11,29,12,30)(13,25,14,26,15,27)(16,23,17,24,18,22)(19,21,20)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$12$:  $D_{6}$
$22$:  $D_{11}$
$44$:  $D_{22}$

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

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $132=2^{2} \cdot 3 \cdot 11$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  132.5
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 3A 6A 11A1 11A2 11A3 11A4 11A5 22A1 22A3 22A5 22A7 22A9 33A1 33A2 33A4 33A5 33A8
Size 1 3 11 33 2 22 2 2 2 2 2 6 6 6 6 6 4 4 4 4 4
2 P 1A 1A 1A 1A 3A 3A 11A2 11A4 11A5 11A3 11A1 11A2 11A5 11A4 11A1 11A3 33A4 33A5 33A8 33A1 33A2
3 P 1A 2A 2B 2C 1A 2B 11A3 11A5 11A2 11A1 11A4 22A5 22A7 22A1 22A3 22A9 11A5 11A2 11A1 11A4 11A3
11 P 1A 2A 2B 2C 3A 6A 11A5 11A1 11A4 11A2 11A3 22A1 22A3 22A9 22A5 22A7 33A1 33A4 33A2 33A8 33A5
Type
132.5.1a R 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
132.5.1b R 1 −1 −1 1 1 −1 1 1 1 1 1 −1 −1 −1 −1 −1 1 1 1 1 1
132.5.1c R 1 −1 1 −1 1 1 1 1 1 1 1 −1 −1 −1 −1 −1 1 1 1 1 1
132.5.1d R 1 1 −1 −1 1 −1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
132.5.2a R 2 0 2 0 −1 −1 2 2 2 2 2 0 0 0 0 0 −1 −1 −1 −1 −1
132.5.2b R 2 0 −2 0 −1 1 2 2 2 2 2 0 0 0 0 0 −1 −1 −1 −1 −1
132.5.2c1 R 2 2 0 0 2 0 ζ11−5+ζ115 ζ11−1+ζ11 ζ11−4+ζ114 ζ11−2+ζ112 ζ11−3+ζ113 ζ11−3+ζ113 ζ11−2+ζ112 ζ11−4+ζ114 ζ11−1+ζ11 ζ11−5+ζ115 ζ11−5+ζ115 ζ11−1+ζ11 ζ11−2+ζ112 ζ11−3+ζ113 ζ11−4+ζ114
132.5.2c2 R 2 2 0 0 2 0 ζ11−4+ζ114 ζ11−3+ζ113 ζ11−1+ζ11 ζ11−5+ζ115 ζ11−2+ζ112 ζ11−2+ζ112 ζ11−5+ζ115 ζ11−1+ζ11 ζ11−3+ζ113 ζ11−4+ζ114 ζ11−4+ζ114 ζ11−3+ζ113 ζ11−5+ζ115 ζ11−2+ζ112 ζ11−1+ζ11
132.5.2c3 R 2 2 0 0 2 0 ζ11−3+ζ113 ζ11−5+ζ115 ζ11−2+ζ112 ζ11−1+ζ11 ζ11−4+ζ114 ζ11−4+ζ114 ζ11−1+ζ11 ζ11−2+ζ112 ζ11−5+ζ115 ζ11−3+ζ113 ζ11−3+ζ113 ζ11−5+ζ115 ζ11−1+ζ11 ζ11−4+ζ114 ζ11−2+ζ112
132.5.2c4 R 2 2 0 0 2 0 ζ11−2+ζ112 ζ11−4+ζ114 ζ11−5+ζ115 ζ11−3+ζ113 ζ11−1+ζ11 ζ11−1+ζ11 ζ11−3+ζ113 ζ11−5+ζ115 ζ11−4+ζ114 ζ11−2+ζ112 ζ11−2+ζ112 ζ11−4+ζ114 ζ11−3+ζ113 ζ11−1+ζ11 ζ11−5+ζ115
132.5.2c5 R 2 2 0 0 2 0 ζ11−1+ζ11 ζ11−2+ζ112 ζ11−3+ζ113 ζ11−4+ζ114 ζ11−5+ζ115 ζ11−5+ζ115 ζ11−4+ζ114 ζ11−3+ζ113 ζ11−2+ζ112 ζ11−1+ζ11 ζ11−1+ζ11 ζ11−2+ζ112 ζ11−4+ζ114 ζ11−5+ζ115 ζ11−3+ζ113
132.5.2d1 R 2 −2 0 0 2 0 ζ11−5+ζ115 ζ11−1+ζ11 ζ11−4+ζ114 ζ11−2+ζ112 ζ11−3+ζ113 −ζ11−3−ζ113 −ζ11−2−ζ112 −ζ11−4−ζ114 −ζ11−1−ζ11 −ζ11−5−ζ115 ζ11−5+ζ115 ζ11−1+ζ11 ζ11−2+ζ112 ζ11−3+ζ113 ζ11−4+ζ114
132.5.2d2 R 2 −2 0 0 2 0 ζ11−4+ζ114 ζ11−3+ζ113 ζ11−1+ζ11 ζ11−5+ζ115 ζ11−2+ζ112 −ζ11−2−ζ112 −ζ11−5−ζ115 −ζ11−1−ζ11 −ζ11−3−ζ113 −ζ11−4−ζ114 ζ11−4+ζ114 ζ11−3+ζ113 ζ11−5+ζ115 ζ11−2+ζ112 ζ11−1+ζ11
132.5.2d3 R 2 −2 0 0 2 0 ζ11−3+ζ113 ζ11−5+ζ115 ζ11−2+ζ112 ζ11−1+ζ11 ζ11−4+ζ114 −ζ11−4−ζ114 −ζ11−1−ζ11 −ζ11−2−ζ112 −ζ11−5−ζ115 −ζ11−3−ζ113 ζ11−3+ζ113 ζ11−5+ζ115 ζ11−1+ζ11 ζ11−4+ζ114 ζ11−2+ζ112
132.5.2d4 R 2 −2 0 0 2 0 ζ11−2+ζ112 ζ11−4+ζ114 ζ11−5+ζ115 ζ11−3+ζ113 ζ11−1+ζ11 −ζ11−1−ζ11 −ζ11−3−ζ113 −ζ11−5−ζ115 −ζ11−4−ζ114 −ζ11−2−ζ112 ζ11−2+ζ112 ζ11−4+ζ114 ζ11−3+ζ113 ζ11−1+ζ11 ζ11−5+ζ115
132.5.2d5 R 2 −2 0 0 2 0 ζ11−1+ζ11 ζ11−2+ζ112 ζ11−3+ζ113 ζ11−4+ζ114 ζ11−5+ζ115 −ζ11−5−ζ115 −ζ11−4−ζ114 −ζ11−3−ζ113 −ζ11−2−ζ112 −ζ11−1−ζ11 ζ11−1+ζ11 ζ11−2+ζ112 ζ11−4+ζ114 ζ11−5+ζ115 ζ11−3+ζ113
132.5.4a1 R 4 0 0 0 −2 0 2ζ11−5+2ζ115 2ζ11−1+2ζ11 2ζ11−4+2ζ114 2ζ11−2+2ζ112 2ζ11−3+2ζ113 0 0 0 0 0 −ζ11−5−ζ115 −ζ11−1−ζ11 −ζ11−2−ζ112 −ζ11−3−ζ113 −ζ11−4−ζ114
132.5.4a2 R 4 0 0 0 −2 0 2ζ11−4+2ζ114 2ζ11−3+2ζ113 2ζ11−1+2ζ11 2ζ11−5+2ζ115 2ζ11−2+2ζ112 0 0 0 0 0 −ζ11−4−ζ114 −ζ11−3−ζ113 −ζ11−5−ζ115 −ζ11−2−ζ112 −ζ11−1−ζ11
132.5.4a3 R 4 0 0 0 −2 0 2ζ11−3+2ζ113 2ζ11−5+2ζ115 2ζ11−2+2ζ112 2ζ11−1+2ζ11 2ζ11−4+2ζ114 0 0 0 0 0 −ζ11−3−ζ113 −ζ11−5−ζ115 −ζ11−1−ζ11 −ζ11−4−ζ114 −ζ11−2−ζ112
132.5.4a4 R 4 0 0 0 −2 0 2ζ11−2+2ζ112 2ζ11−4+2ζ114 2ζ11−5+2ζ115 2ζ11−3+2ζ113 2ζ11−1+2ζ11 0 0 0 0 0 −ζ11−2−ζ112 −ζ11−4−ζ114 −ζ11−3−ζ113 −ζ11−1−ζ11 −ζ11−5−ζ115
132.5.4a5 R 4 0 0 0 −2 0 2ζ11−1+2ζ11 2ζ11−2+2ζ112 2ζ11−3+2ζ113 2ζ11−4+2ζ114 2ζ11−5+2ζ115 0 0 0 0 0 −ζ11−1−ζ11 −ζ11−2−ζ112 −ζ11−4−ζ114 −ζ11−5−ζ115 −ζ11−3−ζ113

magma: CharacterTable(G);