Properties

 Label 15T9 Order $$75$$ n $$15$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_5^2 : C_3$

Related objects

Group action invariants

 Degree $n$ : $15$ Transitive number $t$ : $9$ Group : $C_5^2 : C_3$ CHM label : $[5^{2}]3$ 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) $|\Aut(F/K)|$: $5$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 5: None

Low degree siblings

15T9, 25T6

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

Group invariants

 Order: $75=3 \cdot 5^{2}$ Cyclic: No Abelian: No Solvable: Yes GAP id: [75, 2]
 Character table:  3 1 . . . . 1 1 . . . . 5 2 2 2 2 2 . . 2 2 2 2 1a 5a 5b 5c 5d 3a 3b 5e 5f 5g 5h 2P 1a 5b 5d 5a 5c 3b 3a 5f 5g 5h 5e 3P 1a 5c 5a 5d 5b 1a 1a 5h 5e 5f 5g 5P 1a 1a 1a 1a 1a 3b 3a 1a 1a 1a 1a X.1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 1 1 1 1 D /D 1 1 1 1 X.3 1 1 1 1 1 /D D 1 1 1 1 X.4 3 A /B B /A . . C *C C *C X.5 3 B A /A /B . . *C C *C C X.6 3 /B /A A B . . *C C *C C X.7 3 /A B /B A . . C *C C *C X.8 3 C *C *C C . . B A /B /A X.9 3 C *C *C C . . /B /A B A X.10 3 *C C C *C . . /A B A /B X.11 3 *C C C *C . . A /B /A B A = 2*E(5)+E(5)^3 B = 2*E(5)^3+E(5)^4 C = -E(5)-E(5)^4 = (1-Sqrt(5))/2 = -b5 D = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3