# Properties

 Label 30T10 Degree $30$ Order $60$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $S_3\times D_5$

Show commands: Magma

magma: G := TransitiveGroup(30, 10);

## Group action invariants

 Degree $n$: $30$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $10$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $S_3\times D_5$ 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: (1,27,11,8,21,18)(2,28,12,7,22,17)(3,6,13,16,24,25)(4,5,14,15,23,26)(9,30,19,10,29,20), (1,19,7,26,13)(2,20,8,25,14)(3,11,9,17,15,24,21,29,28,5)(4,12,10,18,16,23,22,30,27,6) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$10$:  $D_{5}$
$12$:  $D_{6}$
$20$:  $D_{10}$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 5: $D_{5}$

Degree 6: $D_{6}$

Degree 10: $D_5$

Degree 15: $D_5\times S_3$

## Low degree siblings

15T7, 30T8, 30T13

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

## Conjugacy classes

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

magma: ConjugacyClasses(G);

## Group invariants

 Order: $60=2^{2} \cdot 3 \cdot 5$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 60.8 magma: IdentifyGroup(G);
 Character table:  2 2 2 2 2 . 1 1 . 1 1 1 1 3 1 . 1 . 1 . 1 1 . 1 1 1 5 1 1 . . 1 1 . 1 1 1 1 1 1a 2a 2b 2c 15a 10a 6a 15b 10b 5a 3a 5b 2P 1a 1a 1a 1a 15b 5a 3a 15a 5b 5b 3a 5a 3P 1a 2a 2b 2c 5a 10b 2b 5b 10a 5b 1a 5a 5P 1a 2a 2b 2c 3a 2a 6a 3a 2a 1a 3a 1a 7P 1a 2a 2b 2c 15b 10b 6a 15a 10a 5b 3a 5a 11P 1a 2a 2b 2c 15a 10a 6a 15b 10b 5a 3a 5b 13P 1a 2a 2b 2c 15b 10b 6a 15a 10a 5b 3a 5a X.1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 1 -1 -1 1 -1 1 1 1 X.3 1 -1 1 -1 1 -1 1 1 -1 1 1 1 X.4 1 1 -1 -1 1 1 -1 1 1 1 1 1 X.5 2 . -2 . -1 . 1 -1 . 2 -1 2 X.6 2 . 2 . -1 . -1 -1 . 2 -1 2 X.7 2 -2 . . A -A . *A -*A *A 2 A X.8 2 -2 . . *A -*A . A -A A 2 *A X.9 2 2 . . A A . *A *A *A 2 A X.10 2 2 . . *A *A . A A A 2 *A X.11 4 . . . -A . . -*A . B -2 *B X.12 4 . . . -*A . . -A . *B -2 B A = E(5)^2+E(5)^3 = (-1-Sqrt(5))/2 = -1-b5 B = 2*E(5)+2*E(5)^4 = -1+Sqrt(5) = 2b5 

magma: CharacterTable(G);