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Properties

Label 33T45
Degree $33$
Order $47520$
Cyclic no
Abelian no
Solvable no
Primitive no
$p$-group no
Group: $S_3\times M_{11}$

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Show commands: Magma

magma: G := TransitiveGroup(33, 45);
 

Group action invariants

Degree $n$:  $33$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $45$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $S_3\times M_{11}$
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:  (2,3)(4,11,17,25)(5,10,18,27)(6,12,16,26)(7,9)(13,22,31,30)(14,24,32,29)(15,23,33,28)(20,21), (1,12,23,4,30,21)(2,10,24,5,28,19)(3,11,22,6,29,20)(7,32,26)(8,33,27)(9,31,25)(13,17,15,16,14,18)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$
$7920$:  $M_{11}$
$15840$:  22T26

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 11: $M_{11}$

Low degree siblings

36T16207

Siblings are shown 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 $ $1$ $1$ $()$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $2$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)(31,32,33)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $3$ $2$ $( 2, 3)( 5, 6)( 7, 9)(10,12)(13,15)(16,18)(20,21)(22,23)(26,27)(28,30)(31,33)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $165$ $2$ $( 1,25)( 2,26)( 3,27)( 4,29)( 5,30)( 6,28)( 7,21)( 8,19)( 9,20)(13,23)(14,24) (15,22)$
$ 6, 6, 6, 6, 3, 3, 3 $ $330$ $6$ $( 1,26, 3,25, 2,27)( 4,30, 6,29, 5,28)( 7,19, 9,21, 8,20)(10,11,12) (13,24,15,23,14,22)(16,17,18)(31,32,33)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $ $495$ $2$ $( 1,25)( 2,27)( 3,26)( 4,29)( 5,28)( 6,30)( 7,20)( 8,19)( 9,21)(10,12)(13,22) (14,24)(15,23)(16,18)(31,33)$
$ 12, 12, 3, 3, 3 $ $1980$ $12$ $( 1, 9,27,19, 2, 7,25,20, 3, 8,26,21)( 4,22,28,14, 5,23,29,15, 6,24,30,13) (10,11,12)(16,17,18)(31,32,33)$
$ 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $990$ $4$ $( 1, 8,25,19)( 2, 9,26,20)( 3, 7,27,21)( 4,24,29,14)( 5,22,30,15)( 6,23,28,13)$
$ 4, 4, 4, 4, 4, 4, 2, 2, 2, 1, 1, 1 $ $2970$ $4$ $( 1, 7,25,21)( 2, 9,26,20)( 3, 8,27,19)( 4,23,29,13)( 5,22,30,15)( 6,24,28,14) (10,11)(16,17)(31,32)$
$ 8, 8, 8, 2, 2, 2, 2, 1 $ $2970$ $8$ $( 1,13, 8, 6,25,23,19,28)( 2,15, 9, 5,26,22,20,30)( 3,14, 7, 4,27,24,21,29) (10,11)(16,32)(17,31)(18,33)$
$ 24, 6, 3 $ $1980$ $24$ $( 1,15, 7, 4,26,23,19,30, 3,14, 9, 6,25,22,21,29, 2,13, 8, 5,27,24,20,28) (10,11,12)(16,32,18,31,17,33)$
$ 8, 8, 8, 2, 2, 2, 1, 1, 1 $ $990$ $8$ $( 1,14, 8, 4,25,24,19,29)( 2,15, 9, 5,26,22,20,30)( 3,13, 7, 6,27,23,21,28) (16,31)(17,32)(18,33)$
$ 24, 6, 3 $ $1980$ $24$ $( 1,30,21,24,26, 6, 8,15, 3,29,20,23,25, 5, 7,14, 2,28,19,22,27, 4, 9,13) (10,11,12)(16,32,18,31,17,33)$
$ 8, 8, 8, 2, 2, 2, 1, 1, 1 $ $990$ $8$ $( 1,29,19,24,25, 4, 8,14)( 2,30,20,22,26, 5, 9,15)( 3,28,21,23,27, 6, 7,13) (16,31)(17,32)(18,33)$
$ 8, 8, 8, 2, 2, 2, 2, 1 $ $2970$ $8$ $( 1,28,19,23,25, 6, 8,13)( 2,30,20,22,26, 5, 9,15)( 3,29,21,24,27, 4, 7,14) (10,11)(16,32)(17,31)(18,33)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $880$ $3$ $( 1,13, 9)( 2,14, 7)( 3,15, 8)( 4,16,26)( 5,17,27)( 6,18,25)(10,30,24) (11,28,22)(12,29,23)(19,21,20)(31,33,32)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1 $ $440$ $3$ $( 1,14, 8)( 2,15, 9)( 3,13, 7)( 4,17,25)( 5,18,26)( 6,16,27)(10,28,23) (11,29,24)(12,30,22)$
$ 6, 6, 6, 3, 3, 3, 2, 2, 1, 1 $ $1320$ $6$ $( 1,15, 8, 2,14, 9)( 3,13, 7)( 4,18,25, 5,17,26)( 6,16,27)(10,28,23) (11,30,24,12,29,22)(19,20)(32,33)$
$ 6, 6, 6, 6, 3, 2, 2, 2 $ $3960$ $6$ $( 1, 9,14, 2, 8,15)( 3, 7,13)( 4,30,17,22,25,12)( 5,29,18,24,26,11) ( 6,28,16,23,27,10)(19,33)(20,32)(21,31)$
$ 6, 6, 6, 6, 3, 3, 3 $ $2640$ $6$ $( 1, 7,15)( 2, 8,13)( 3, 9,14)( 4,28,18,24,27,12)( 5,29,16,22,25,10) ( 6,30,17,23,26,11)(19,31,20,32,21,33)$
$ 6, 6, 6, 3, 3, 3, 2, 2, 2 $ $1320$ $6$ $( 1, 8,14)( 2, 9,15)( 3, 7,13)( 4,29,17,24,25,11)( 5,30,18,22,26,12) ( 6,28,16,23,27,10)(19,32)(20,33)(21,31)$
$ 10, 10, 5, 5, 2, 1 $ $4752$ $10$ $( 2, 3)( 4,11,32,14,24)( 5,10,33,13,22, 6,12,31,15,23)( 7,18,21,30,27, 9,16, 20,28,26)( 8,17,19,29,25)$
$ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ $1584$ $5$ $( 4,11,32,14,24)( 5,12,33,15,22)( 6,10,31,13,23)( 7,16,21,28,27) ( 8,17,19,29,25)( 9,18,20,30,26)$
$ 15, 15, 3 $ $3168$ $15$ $( 1, 2, 3)( 4,12,31,14,22, 6,11,33,13,24, 5,10,32,15,23)( 7,17,20,28,25, 9,16, 19,30,27, 8,18,21,29,26)$
$ 33 $ $1440$ $33$ $( 1,22,13, 4,12,27,32,20,16, 8,30, 3,24,15, 6,11,26,31,19,18, 7,29, 2,23,14, 5,10,25,33,21,17, 9,28)$
$ 11, 11, 11 $ $720$ $11$ $( 1,24,14, 4,11,25,32,19,17, 8,29)( 2,22,15, 5,12,26,33,20,18, 9,30) ( 3,23,13, 6,10,27,31,21,16, 7,28)$
$ 22, 11 $ $2160$ $22$ $( 1,23,14, 6,11,27,32,21,17, 7,29, 3,24,13, 4,10,25,31,19,16, 8,28) ( 2,22,15, 5,12,26,33,20,18, 9,30)$
$ 33 $ $1440$ $33$ $( 1,30, 7,17,20,31,25,12, 6,14,22, 3,29, 9,16,19,33,27,11, 5,13,24, 2,28, 8, 18,21,32,26,10, 4,15,23)$
$ 11, 11, 11 $ $720$ $11$ $( 1,29, 8,17,19,32,25,11, 4,14,24)( 2,30, 9,18,20,33,26,12, 5,15,22) ( 3,28, 7,16,21,31,27,10, 6,13,23)$
$ 22, 11 $ $2160$ $22$ $( 1,28, 8,16,19,31,25,10, 4,13,24, 3,29, 7,17,21,32,27,11, 6,14,23) ( 2,30, 9,18,20,33,26,12, 5,15,22)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $47520=2^{5} \cdot 3^{3} \cdot 5 \cdot 11$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  no
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  47520.a
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);