# Properties

 Label 15T12 Order $$150$$ n $$15$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $(C_5^2 : C_3):C_2$

# Related objects

## Group action invariants

 Degree $n$ : $15$ Transitive number $t$ : $12$ Group : $(C_5^2 : C_3):C_2$ CHM label : $[5^{2}:2]3$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,4)(2,8)(3,12)(6,9)(7,13)(11,14), (1,13,10,7,4)(2,5,8,11,14), (1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15) $|\Aut(F/K)|$: $1$

## 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 3: $C_3$

Degree 5: None

## Low degree siblings

15T12, 25T15, 30T35 x 2

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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $2, 2, 2, 2, 2, 2, 1, 1, 1$ $25$ $2$ $( 3, 6)( 4,13)( 5,14)( 7,10)( 8,11)( 9,15)$ $5, 5, 1, 1, 1, 1, 1$ $6$ $5$ $( 2, 5, 8,11,14)( 3,15,12, 9, 6)$ $5, 5, 1, 1, 1, 1, 1$ $6$ $5$ $( 2, 8,14, 5,11)( 3,12, 6,15, 9)$ $3, 3, 3, 3, 3$ $25$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)$ $6, 6, 3$ $25$ $6$ $( 1, 2, 3, 4,14, 6)( 5,15, 7,11, 9,13)( 8,12,10)$ $3, 3, 3, 3, 3$ $25$ $3$ $( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)$ $6, 6, 3$ $25$ $6$ $( 1, 3, 5,13, 6, 2)( 4,15, 8,10, 9,14)( 7,12,11)$ $5, 5, 5$ $6$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,12, 6,15, 9)$ $5, 5, 5$ $6$ $5$ $( 1, 4, 7,10,13)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$

## Group invariants

 Order: $150=2 \cdot 3 \cdot 5^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [150, 6]
 Character table:  2 1 1 . . 1 1 1 1 . . 3 1 1 . . 1 1 1 1 . . 5 2 . 2 2 . . . . 2 2 1a 2a 5a 5b 3a 6a 3b 6b 5c 5d 2P 1a 1a 5b 5a 3b 3b 3a 3a 5d 5c 3P 1a 2a 5b 5a 1a 2a 1a 2a 5d 5c 5P 1a 2a 1a 1a 3b 6b 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)^2 = (-1-Sqrt(-3))/2 = -1-b3