Properties

 Label 15T14 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 $t$ : $14$ Group : $(C_5^2 : C_3):C_2$ CHM label : $5^{2}:2[1/2]S(3)$ Parity: $-1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (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), (1,14)(2,13)(3,12)(4,11)(5,10)(6,9)(7,8) $|\Aut(F/K)|$: $1$

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 3: $S_3$

Degree 5: None

Low degree siblings

15T13, 25T16, 30T37, 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 $1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1$ $1$ $1$ $()$ $10, 2, 2, 1$ $15$ $10$ $( 2, 3, 5,15, 8,12,11, 9,14, 6)( 4,13)( 7,10)$ $5, 5, 1, 1, 1, 1, 1$ $3$ $5$ $( 2, 5, 8,11,14)( 3,15,12, 9, 6)$ $10, 2, 2, 1$ $15$ $10$ $( 2, 6,14, 9,11,12, 8,15, 5, 3)( 4,13)( 7,10)$ $5, 5, 1, 1, 1, 1, 1$ $3$ $5$ $( 2, 8,14, 5,11)( 3,12, 6,15, 9)$ $10, 2, 2, 1$ $15$ $10$ $( 2, 9, 8, 3,14,12, 5, 6,11,15)( 4,13)( 7,10)$ $5, 5, 1, 1, 1, 1, 1$ $3$ $5$ $( 2,11, 5,14, 8)( 3, 9,15, 6,12)$ $2, 2, 2, 2, 2, 2, 2, 1$ $15$ $2$ $( 2,12)( 3,11)( 4,13)( 5, 9)( 6, 8)( 7,10)(14,15)$ $5, 5, 1, 1, 1, 1, 1$ $3$ $5$ $( 2,14,11, 8, 5)( 3, 6, 9,12,15)$ $10, 2, 2, 1$ $15$ $10$ $( 2,15,11, 6, 5,12,14, 3, 8, 9)( 4,13)( 7,10)$ $3, 3, 3, 3, 3$ $50$ $3$ $( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)$ $5, 5, 5$ $6$ $5$ $( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3,12, 6,15, 9)$ $5, 5, 5$ $6$ $5$ $( 1, 4, 7,10,13)( 2, 8,14, 5,11)( 3, 9,15, 6,12)$

Group invariants

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