# Properties

 Label 30T37 Degree $30$ Order $150$ Cyclic no Abelian no Solvable yes Primitive no $p$-group no Group: $C_5^2:S_3$

## Group action invariants

 Degree $n$: $30$ Transitive number $t$: $37$ Group: $C_5^2:S_3$ Parity: $-1$ Primitive: no Nilpotency class: $-1$ (not nilpotent) $|\Aut(F/K)|$: $10$ Generators: (1,4,25,10,19,15,13,21,7,28)(2,3,26,9,20,16,14,22,8,27)(5,17)(6,18)(11,12)(23,29)(24,30), (1,11,22)(2,12,21)(3,13,23)(4,14,24)(5,15,26)(6,16,25)(7,17,27)(8,18,28)(9,19,30)(10,20,29)

## Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$
$6$:  $S_3$

Resolvents shown for degrees $\leq 47$

## Subfields

Degree 2: $C_2$

Degree 3: $S_3$

Degree 5: None

Degree 6: $S_3$

Degree 10: None

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

## Low degree siblings

15T13, 15T14, 25T16, 30T38

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $5$ $( 3, 9,16,22,27)( 4,10,15,21,28)( 5,29,24,18,12)( 6,30,23,17,11)$ $5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $5$ $( 3,16,27, 9,22)( 4,15,28,10,21)( 5,24,12,29,18)( 6,23,11,30,17)$ $5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $5$ $( 3,22, 9,27,16)( 4,21,10,28,15)( 5,18,29,12,24)( 6,17,30,11,23)$ $5, 5, 5, 5, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $3$ $5$ $( 3,27,22,16, 9)( 4,28,21,15,10)( 5,12,18,24,29)( 6,11,17,23,30)$ $10, 10, 2, 2, 2, 2, 2$ $15$ $10$ $( 1, 2)( 3, 5, 9,29,16,24,22,18,27,12)( 4, 6,10,30,15,23,21,17,28,11)( 7,26) ( 8,25)(13,20)(14,19)$ $10, 10, 2, 2, 2, 2, 2$ $15$ $10$ $( 1, 2)( 3,12,27,18,22,24,16,29, 9, 5)( 4,11,28,17,21,23,15,30,10, 6)( 7,26) ( 8,25)(13,20)(14,19)$ $10, 10, 2, 2, 2, 2, 2$ $15$ $10$ $( 1, 2)( 3,18,16, 5,27,24, 9,12,22,29)( 4,17,15, 6,28,23,10,11,21,30)( 7,26) ( 8,25)(13,20)(14,19)$ $2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2$ $15$ $2$ $( 1, 2)( 3,24)( 4,23)( 5,22)( 6,21)( 7,26)( 8,25)( 9,18)(10,17)(11,15)(12,16) (13,20)(14,19)(27,29)(28,30)$ $10, 10, 2, 2, 2, 2, 2$ $15$ $10$ $( 1, 2)( 3,29,22,12, 9,24,27, 5,16,18)( 4,30,21,11,10,23,28, 6,15,17)( 7,26) ( 8,25)(13,20)(14,19)$ $3, 3, 3, 3, 3, 3, 3, 3, 3, 3$ $50$ $3$ $( 1, 3, 6)( 2, 4, 5)( 7, 9,11)( 8,10,12)(13,16,17)(14,15,18)(19,22,23) (20,21,24)(25,27,30)(26,28,29)$ $5, 5, 5, 5, 5, 5$ $6$ $5$ $( 1, 7,13,19,25)( 2, 8,14,20,26)( 3, 9,16,22,27)( 4,10,15,21,28) ( 5,24,12,29,18)( 6,23,11,30,17)$ $5, 5, 5, 5, 5, 5$ $6$ $5$ $( 1, 7,13,19,25)( 2, 8,14,20,26)( 3,16,27, 9,22)( 4,15,28,10,21) ( 5,18,29,12,24)( 6,17,30,11,23)$

## Group invariants

 Order: $150=2 \cdot 3 \cdot 5^{2}$ Cyclic: no Abelian: no Solvable: yes GAP id: [150, 5]
 Character table:  2 1 1 1 1 1 1 1 1 1 1 . . . 3 1 . . . . . . . . . 1 . . 5 2 2 2 2 2 1 1 1 1 1 . 2 2 1a 5a 5b 5c 5d 10a 10b 10c 2a 10d 3a 5e 5f 2P 1a 5b 5d 5a 5c 5a 5d 5b 1a 5c 3a 5f 5e 3P 1a 5c 5a 5d 5b 10d 10c 10a 2a 10b 1a 5f 5e 5P 1a 1a 1a 1a 1a 2a 2a 2a 2a 2a 3a 1a 1a 7P 1a 5b 5d 5a 5c 10c 10d 10b 2a 10a 3a 5f 5e X.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 X.3 2 2 2 2 2 . . . . . -1 2 2 X.4 3 A /B B /A D /D /E -1 E . F *F X.5 3 B A /A /B E /E D -1 /D . *F F X.6 3 /A B /B A /D D E -1 /E . F *F X.7 3 /B /A A B /E E /D -1 D . *F F X.8 3 A /B B /A -D -/D -/E 1 -E . F *F X.9 3 B A /A /B -E -/E -D 1 -/D . *F F X.10 3 /A B /B A -/D -D -E 1 -/E . F *F X.11 3 /B /A A B -/E -E -/D 1 -D . *F F X.12 6 C *C *C C . . . . . . G *G X.13 6 *C C C *C . . . . . . *G G A = 2*E(5)^3+E(5)^4 B = E(5)^2+2*E(5)^4 C = -2*E(5)-2*E(5)^4 = 1-Sqrt(5) = 1-r5 D = -E(5)^2 E = -E(5) F = -E(5)^2-E(5)^3 = (1+Sqrt(5))/2 = 1+b5 G = E(5)+2*E(5)^2+2*E(5)^3+E(5)^4 = (-3-Sqrt(5))/2 = -2-b5