# Properties

 Label 30T20 Degree $30$ Order $120$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $D_5\times A_4$

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magma: G := TransitiveGroup(30, 20);

## Group action invariants

 Degree $n$: $30$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $20$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $D_5\times A_4$ 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: (1,4,5,8,10)(2,3,6,7,9)(11,14,16,17,19,12,13,15,18,20)(21,24,26,27,30,22,23,25,28,29), (1,13,27,2,14,28)(3,12,30,10,16,25)(4,11,29,9,15,26)(5,19,22,7,17,23)(6,20,21,8,18,24) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$
$10$:  $D_{5}$
$12$:  $A_4$
$24$:  $A_4\times C_2$
$30$:  $D_5\times C_3$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: None

Degree 3: $C_3$

Degree 5: $D_{5}$

Degree 6: $A_4\times C_2$

Degree 10: None

Degree 15: $D_5\times C_3$

## Low degree siblings

20T37, 30T28, 40T65

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

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: 120.39 magma: IdentifyGroup(G);
 Character table:  2 3 3 3 3 2 2 2 2 . 1 1 . . 1 . 1 3 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 1a 2a 2b 2c 10a 5a 10b 5b 15a 6a 3a 15b 15c 6b 15d 3b 2P 1a 1a 1a 1a 5b 5b 5a 5a 15c 3b 3b 15d 15a 3a 15b 3a 3P 1a 2a 2b 2c 10b 5b 10a 5a 5b 2c 1a 5a 5a 2c 5b 1a 5P 1a 2a 2b 2c 2a 1a 2a 1a 3b 6b 3b 3b 3a 6a 3a 3a 7P 1a 2a 2b 2c 10b 5b 10a 5a 15b 6a 3a 15a 15d 6b 15c 3b 11P 1a 2a 2b 2c 10a 5a 10b 5b 15d 6b 3b 15c 15b 6a 15a 3a 13P 1a 2a 2b 2c 10b 5b 10a 5a 15b 6a 3a 15a 15d 6b 15c 3b X.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 X.3 1 1 -1 -1 1 1 1 1 C -C C C /C -/C /C /C X.4 1 1 -1 -1 1 1 1 1 /C -/C /C /C C -C C C X.5 1 1 1 1 1 1 1 1 C C C C /C /C /C /C X.6 1 1 1 1 1 1 1 1 /C /C /C /C C C C C X.7 2 2 . . A A *A *A A . 2 *A *A . A 2 X.8 2 2 . . *A *A A A *A . 2 A A . *A 2 X.9 2 2 . . A A *A *A D . F E /E . /D /F X.10 2 2 . . A A *A *A /D . /F /E E . D F X.11 2 2 . . *A *A A A E . F D /D . /E /F X.12 2 2 . . *A *A A A /E . /F /D D . E F X.13 3 -1 -1 3 -1 3 -1 3 . . . . . . . . X.14 3 -1 1 -3 -1 3 -1 3 . . . . . . . . X.15 6 -2 . . -A B -*A *B . . . . . . . . X.16 6 -2 . . -*A *B -A B . . . . . . . . A = E(5)+E(5)^4 = (-1+Sqrt(5))/2 = b5 B = 3*E(5)+3*E(5)^4 = (-3+3*Sqrt(5))/2 = 3b5 C = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 D = E(15)^7+E(15)^13 E = E(15)+E(15)^4 F = 2*E(3)^2 = -1-Sqrt(-3) = -1-i3 

magma: CharacterTable(G);