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Properties

Label 30T33
Degree $30$
Order $120$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_5\times S_4$

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Show commands: Magma

magma: G := TransitiveGroup(30, 33);
 

Group action invariants

Degree $n$:  $30$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $33$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_5\times S_4$
Parity:  $1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
Nilpotency class:  $-1$ (not nilpotent)
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $10$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,29,6,23,9,28,3,21,8,26)(2,30,5,24,10,27,4,22,7,25)(11,14,15,17,19)(12,13,16,18,20), (1,27,13)(2,28,14)(3,30,16)(4,29,15)(5,21,17)(6,22,18)(7,23,19)(8,24,20)(9,25,12)(10,26,11)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$5$:  $C_5$
$6$:  $S_3$
$10$:  $C_{10}$
$24$:  $S_4$
$30$:  $S_3 \times C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $S_3$

Degree 5: $C_5$

Degree 6: $S_4$

Degree 10: None

Degree 15: $S_3 \times C_5$

Low degree siblings

20T34, 30T34, 40T64

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $120=2^{3} \cdot 3 \cdot 5$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Label:  120.37
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);