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Properties

Label 30T18
Degree $30$
Order $120$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{10}\times A_4$

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Show commands: Magma

magma: G := TransitiveGroup(30, 18);
 

Group action invariants

Degree $n$:  $30$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $18$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{10}\times A_4$
Parity:  $-1$
magma: IsEven(G);
 
Primitive:  no
magma: IsPrimitive(G);
 
magma: NilpotencyClass(G);
 
$\card{\Aut(F/K)}$:  $10$
magma: Order(Centralizer(SymmetricGroup(n), G));
 
Generators:  (1,16,29,5,19,23,9,14,27,3,18,22,7,11,25,2,15,30,6,20,24,10,13,28,4,17,21,8,12,26), (1,2)(3,4)(5,6)(7,8)(9,10)(21,22)(23,24)(25,26)(27,28)(29,30)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$5$:  $C_5$
$6$:  $C_6$
$10$:  $C_{10}$
$12$:  $A_4$
$15$:  $C_{15}$
$24$:  $A_4\times C_2$
$30$:  $C_{30}$
$60$:  20T14

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 5: $C_5$

Degree 6: $A_4\times C_2$

Degree 10: None

Degree 15: $C_{15}$

Low degree siblings

40T59

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

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);
 
Nilpotency class:   not nilpotent
Label:  120.43
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);