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Properties

Label 33T21
Degree $33$
Order $3630$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5^4:D_4$

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Show commands: Magma

magma: G := TransitiveGroup(33, 21);
 

Group action invariants

Degree $n$:  $33$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $21$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5^4:D_4$
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:  (1,6,9,2,11)(3,5,4,10,7)(12,29,17,30,21,33,22,31,14,25)(13,27,20,24,19,26,16,32,18,28)(15,23), (1,19,25,3,21,27,2,20,26,8,15,32,5,12,29)(4,22,28,7,14,31,11,18,24,9,16,33,10,17,23)(6,13,30)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$6$:  $S_3$
$10$:  $C_{10}$
$30$:  $S_3 \times C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 11: None

Low degree siblings

33T20

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 $ $1$ $1$ $()$
$ 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $30$ $11$ $( 1, 6,11, 5,10, 4, 9, 3, 8, 2, 7)(23,31,28,25,33,30,27,24,32,29,26)$
$ 11, 11, 11 $ $30$ $11$ $( 1, 6,11, 5,10, 4, 9, 3, 8, 2, 7)(12,22,21,20,19,18,17,16,15,14,13) (23,24,25,26,27,28,29,30,31,32,33)$
$ 11, 11, 11 $ $30$ $11$ $( 1,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(12,22,21,20,19,18,17,16,15,14,13) (23,32,30,28,26,24,33,31,29,27,25)$
$ 11, 11, 11 $ $15$ $11$ $( 1, 8, 4,11, 7, 3,10, 6, 2, 9, 5)(12,22,21,20,19,18,17,16,15,14,13) (23,25,27,29,31,33,24,26,28,30,32)$
$ 11, 11, 11 $ $15$ $11$ $( 1, 6,11, 5,10, 4, 9, 3, 8, 2, 7)(12,21,19,17,15,13,22,20,18,16,14) (23,28,33,27,32,26,31,25,30,24,29)$
$ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ $121$ $5$ $( 2, 4,10, 6, 5)( 3, 7, 8,11, 9)(13,15,21,17,16)(14,18,19,22,20) (23,28,32,33,25)(24,31,30,27,29)$
$ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ $121$ $5$ $( 2,10, 5, 4, 6)( 3, 8, 9, 7,11)(13,21,16,15,17)(14,19,20,18,22) (23,32,25,28,33)(24,30,29,31,27)$
$ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ $121$ $5$ $( 2, 5, 6,10, 4)( 3, 9,11, 8, 7)(13,16,17,21,15)(14,20,22,19,18) (23,25,33,32,28)(24,29,27,30,31)$
$ 5, 5, 5, 5, 5, 5, 1, 1, 1 $ $121$ $5$ $( 2, 6, 4, 5,10)( 3,11, 7, 9, 8)(13,17,15,16,21)(14,22,18,20,19) (23,33,28,25,32)(24,27,31,29,30)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $ $242$ $3$ $( 1,31,16)( 2,25,20)( 3,30,13)( 4,24,17)( 5,29,21)( 6,23,14)( 7,28,18) ( 8,33,22)( 9,27,15)(10,32,19)(11,26,12)$
$ 15, 15, 3 $ $242$ $15$ $( 1,30,15, 3,27,21, 2,23,18, 8,25,14, 5,24,16)( 4,31,13, 7,32,22,11,26,12, 9, 29,17,10,33,20)( 6,28,19)$
$ 15, 15, 3 $ $242$ $15$ $( 1,27,17, 6,32,20,10,25,18,11,26,12, 3,29,16)( 2,28,22, 9,24,13, 8,23,19, 5, 31,15, 7,33,14)( 4,30,21)$
$ 15, 15, 3 $ $242$ $15$ $( 1,24,21, 6,25,22, 7,23,20, 5,27,13, 9,30,16)( 2,33,19, 4,29,15,11,26,12, 8, 32,18, 3,31,17)(10,28,14)$
$ 15, 15, 3 $ $242$ $15$ $( 1,29,13,11,26,12, 7,25,19, 2,32,14, 4,27,16)( 3,24,15, 8,28,20, 6,33,18, 9, 31,21,10,23,22)( 5,30,17)$
$ 10, 10, 5, 5, 2, 1 $ $363$ $10$ $( 2, 6, 4, 5,10)( 3,11, 7, 9, 8)(12,32,18,31,14,28,13,30,21,25) (15,26,16,24,19,29,17,33,22,23)(20,27)$
$ 10, 10, 5, 5, 2, 1 $ $363$ $10$ $( 2, 5, 6,10, 4)( 3, 9,11, 8, 7)(12,33,20,29,16,31,18,30,17,25) (13,27,14,32,19,24,22,28,15,26)(21,23)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $33$ $2$ $(12,25)(13,29)(14,33)(15,26)(16,30)(17,23)(18,27)(19,31)(20,24)(21,28)(22,32)$
$ 22, 11 $ $165$ $22$ $( 1, 6,11, 5,10, 4, 9, 3, 8, 2, 7)(12,33,14,30,16,27,18,24,20,32,22,29,13,26, 15,23,17,31,19,28,21,25)$
$ 22, 11 $ $165$ $22$ $( 1,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(12,30,16,24,20,29,13,23,17,28,21,33,14,27, 18,32,22,26,15,31,19,25)$
$ 10, 10, 5, 5, 2, 1 $ $363$ $10$ $( 2,10, 5, 4, 6)( 3, 8, 9, 7,11)(12,28,16,29,21,33,19,27,22,25) (13,31,20,30,15,26,17,32,14,23)(18,24)$
$ 10, 10, 5, 5, 2, 1 $ $363$ $10$ $( 2, 4,10, 6, 5)( 3, 7, 8,11, 9)(12,23,16,27,19,30,13,24,14,25) (15,26,21,32,20,31,22,33,18,29)(17,28)$

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $3630=2 \cdot 3 \cdot 5 \cdot 11^{2}$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  3630.p
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);