Properties

 Label 15T6 Order $$60$$ n $$15$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_{15} : C_4$

Related objects

Group action invariants

 Degree $n$ : $15$ Transitive number $t$ : $6$ Group : $C_{15} : C_4$ CHM label : $F(5)[1/2]S(3)$ Parity: $1$ Primitive: No Nilpotency class: $-1$ (not nilpotent) Generators: (1,2,4,8)(3,6,12,9)(5,10)(7,14,13,11), (1,4)(2,8)(3,12)(6,9)(7,13)(11,14), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15) $|\Aut(F/K)|$: $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
6:  $S_3$
12:  $C_3 : C_4$
20:  $F_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $S_3$

Degree 5: $F_5$

Low degree siblings

30T6

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

Group invariants

 Order: $60=2^{2} \cdot 3 \cdot 5$ Cyclic: No Abelian: No Solvable: Yes GAP id: [60, 7]
 Character table:  2 2 2 2 2 . 1 . 1 . 3 1 . 1 . 1 1 1 1 1 5 1 . . . 1 . 1 1 1 1a 4a 2a 4b 15a 6a 5a 3a 15b 2P 1a 2a 1a 2a 15a 3a 5a 3a 15b 3P 1a 4b 2a 4a 5a 2a 5a 1a 5a 5P 1a 4a 2a 4b 3a 6a 1a 3a 3a 7P 1a 4b 2a 4a 15b 6a 5a 3a 15a 11P 1a 4b 2a 4a 15b 6a 5a 3a 15a 13P 1a 4a 2a 4b 15b 6a 5a 3a 15a X.1 1 1 1 1 1 1 1 1 1 X.2 1 -1 1 -1 1 1 1 1 1 X.3 1 A -1 -A 1 -1 1 1 1 X.4 1 -A -1 A 1 -1 1 1 1 X.5 2 . -2 . -1 1 2 -1 -1 X.6 2 . 2 . -1 -1 2 -1 -1 X.7 4 . . . -1 . -1 4 -1 X.8 4 . . . B . -1 -2 /B X.9 4 . . . /B . -1 -2 B A = -E(4) = -Sqrt(-1) = -i B = E(15)^7+E(15)^11+E(15)^13+E(15)^14 = (1-Sqrt(-15))/2 = -b15