# Properties

 Label 30T35 Degree $30$ Order $150$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_5^2:C_6$

Show commands: Magma

magma: G := TransitiveGroup(30, 35);

## Group action invariants

 Degree $n$: $30$ magma: t, n := TransitiveGroupIdentification(G); n; Transitive number $t$: $35$ magma: t, n := TransitiveGroupIdentification(G); t; Group: $C_5^2:C_6$ 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: (3,10,15,21,27)(4,9,16,22,28)(5,29,23,17,11)(6,30,24,18,12), (1,18,22,8,11,27)(2,17,21,7,12,28)(3,25,24,16,14,5)(4,26,23,15,13,6)(9,20,29,10,19,30) magma: Generators(G);

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$3$:  $C_3$
$6$:  $C_6$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: $C_3$

Degree 5: None

Degree 6: $C_6$

Degree 10: None

Degree 15: $(C_5^2 : C_3):C_2$

## Low degree siblings

15T12 x 2, 25T15, 30T35

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

magma: ConjugacyClasses(G);

## Group invariants

 Order: $150=2 \cdot 3 \cdot 5^{2}$ magma: Order(G); Cyclic: no magma: IsCyclic(G); Abelian: no magma: IsAbelian(G); Solvable: yes magma: IsSolvable(G); Label: 150.6 magma: IdentifyGroup(G);
 Character table:  2 1 . . 1 1 1 1 1 . . 3 1 . . 1 1 1 1 1 . . 5 2 2 2 . . . . . 2 2 1a 5a 5b 2a 6a 3a 3b 6b 5c 5d 2P 1a 5b 5a 1a 3b 3b 3a 3a 5d 5c 3P 1a 5b 5a 2a 2a 1a 1a 2a 5d 5c 5P 1a 1a 1a 2a 6b 3b 3a 6a 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 -1 -1 1 1 -1 1 1 X.3 1 1 1 -1 C -C -/C /C 1 1 X.4 1 1 1 -1 /C -/C -C C 1 1 X.5 1 1 1 1 -/C -/C -C -C 1 1 X.6 1 1 1 1 -C -C -/C -/C 1 1 X.7 6 A *A . . . . . B *B X.8 6 *A A . . . . . *B B X.9 6 B *B . . . . . *A A X.10 6 *B B . . . . . A *A A = -2*E(5)-2*E(5)^4 = 1-Sqrt(5) = 1-r5 B = E(5)+2*E(5)^2+2*E(5)^3+E(5)^4 = (-3-Sqrt(5))/2 = -2-b5 C = -E(3) = (1-Sqrt(-3))/2 = -b3 

magma: CharacterTable(G);