# Properties

 Label 30T14 Degree $30$ Order $60$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $D_{30}$

# Related objects

Show commands: Magma

magma: G := TransitiveGroup(30, 14);

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$6$:  $S_3$
$10$:  $D_{5}$
$12$:  $D_{6}$
$20$:  $D_{10}$
$30$:  $D_{15}$

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: $D_{10}$

Degree 15: $D_{15}$

## Low degree siblings

30T14

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

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: 60.12 magma: IdentifyGroup(G);
 Character table:  2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 3 1 . 1 . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5 1 . 1 . 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1a 2a 2b 2c 30a 15a 15b 30b 5a 10a 30c 15c 3a 6a 10b 5b 15d 30d 2P 1a 1a 1a 1a 15b 15b 15c 15c 5b 5b 15d 15d 3a 3a 5a 5a 15a 15a 3P 1a 2a 2b 2c 10a 5a 5b 10b 5b 10b 10a 5a 1a 2b 10a 5a 5b 10b 5P 1a 2a 2b 2c 6a 3a 3a 6a 1a 2b 6a 3a 3a 6a 2b 1a 3a 6a 7P 1a 2a 2b 2c 30d 15d 15a 30a 5b 10b 30b 15b 3a 6a 10a 5a 15c 30c 11P 1a 2a 2b 2c 30c 15c 15d 30d 5a 10a 30a 15a 3a 6a 10b 5b 15b 30b 13P 1a 2a 2b 2c 30b 15b 15c 30c 5b 10b 30d 15d 3a 6a 10a 5a 15a 30a 17P 1a 2a 2b 2c 30b 15b 15c 30c 5b 10b 30d 15d 3a 6a 10a 5a 15a 30a 19P 1a 2a 2b 2c 30c 15c 15d 30d 5a 10a 30a 15a 3a 6a 10b 5b 15b 30b 23P 1a 2a 2b 2c 30d 15d 15a 30a 5b 10b 30b 15b 3a 6a 10a 5a 15c 30c 29P 1a 2a 2b 2c 30a 15a 15b 30b 5a 10a 30c 15c 3a 6a 10b 5b 15d 30d X.1 1 1 1 1 1 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 1 -1 -1 1 1 -1 X.3 1 -1 1 -1 1 1 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 1 -1 -1 1 1 -1 X.5 2 . 2 . -1 -1 -1 -1 2 2 -1 -1 -1 -1 2 2 -1 -1 X.6 2 . -2 . 1 -1 -1 1 2 -2 1 -1 -1 1 -2 2 -1 1 X.7 2 . -2 . A -A -*A *A -*A *A A -A 2 -2 A -A -*A *A X.8 2 . -2 . *A -*A -A A -A A *A -*A 2 -2 *A -*A -A A X.9 2 . 2 . -*A -*A -A -A -A -A -*A -*A 2 2 -*A -*A -A -A X.10 2 . 2 . -A -A -*A -*A -*A -*A -A -A 2 2 -A -A -*A -*A X.11 2 . -2 . B -B -C C -A A D -D -1 1 *A -*A -E E X.12 2 . -2 . C -C -D D -*A *A E -E -1 1 A -A -B B X.13 2 . -2 . D -D -E E -A A B -B -1 1 *A -*A -C C X.14 2 . -2 . E -E -B B -*A *A C -C -1 1 A -A -D D X.15 2 . 2 . -E -E -B -B -*A -*A -C -C -1 -1 -A -A -D -D X.16 2 . 2 . -D -D -E -E -A -A -B -B -1 -1 -*A -*A -C -C X.17 2 . 2 . -C -C -D -D -*A -*A -E -E -1 -1 -A -A -B -B X.18 2 . 2 . -B -B -C -C -A -A -D -D -1 -1 -*A -*A -E -E A = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5 B = -E(15)-E(15)^14 C = -E(15)^2-E(15)^13 D = -E(15)^4-E(15)^11 E = -E(15)^7-E(15)^8 

magma: CharacterTable(G);