Galois Group: 15T24

• Label: 15T24
• Order: \(360\)
• n: \(15\)
• Cyclic: No
• Abelian: No
• Solvable: No
• Primitive: No
• $p$-group: No
• Group: $S_5 \times C_3$

Group action invariants

Degree $n$ :		$15$
Transitive number $t$ :		$24$
Group :		$S_5 \times C_3$
CHM label :		$S(5)[x]3$
Parity:		$-1$
Primitive:		No
Nilpotency class:		$-1$ (not nilpotent)
Generators:		(1,4)(6,9)(11,14), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15)
$|\Aut(F/K)|$:		$3$

Low degree resolvents

|G/N|	Galois groups for stem field(s)
2:	$C_2$
3:	$C_3$
6:	$C_6$
120:	$S_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 5: $S_5$

Low degree siblings

18T144, 30T90, 30T98, 30T103, 36T550, 45T44

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, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$10$	$2$	$( 4,10)( 5,14)( 9,15)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $	$20$	$3$	$( 3, 9,15)( 4,10,13)( 5, 8,14)$
$ 4, 4, 4, 1, 1, 1 $	$30$	$4$	$( 2, 5, 8,14)( 3, 9,12,15)( 4, 7,10,13)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $	$15$	$2$	$( 2, 5)( 3, 9)( 4,13)( 7,10)( 8,14)(12,15)$
$ 6, 3, 3, 3 $	$20$	$6$	$( 1, 2, 3)( 4, 5, 9,10,14,15)( 6, 7, 8)(11,12,13)$
$ 3, 3, 3, 3, 3 $	$20$	$3$	$( 1, 2, 3)( 4,14, 9)( 5,15,10)( 6, 7, 8)(11,12,13)$
$ 15 $	$24$	$15$	$( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11,12,13,14,15)$
$ 12, 3 $	$30$	$12$	$( 1, 2, 3, 4,11,12,13,14, 6, 7, 8, 9)( 5,15,10)$
$ 6, 6, 3 $	$15$	$6$	$( 1, 2, 6, 7,11,12)( 3, 4, 8, 9,13,14)( 5,15,10)$
$ 6, 3, 3, 3 $	$10$	$6$	$( 1, 2, 6, 7,11,12)( 3,13, 8)( 4,14, 9)( 5,15,10)$
$ 3, 3, 3, 3, 3 $	$20$	$3$	$( 1, 3, 2)( 4, 9,14)( 5,10,15)( 6, 8, 7)(11,13,12)$
$ 6, 3, 3, 3 $	$20$	$6$	$( 1, 3, 2)( 4,15,14,10, 9, 5)( 6, 8, 7)(11,13,12)$
$ 12, 3 $	$30$	$12$	$( 1, 3, 5, 7, 6, 8,10,12,11,13,15, 2)( 4, 9,14)$
$ 15 $	$24$	$15$	$( 1, 3, 5, 4,12,11,13,15,14, 7, 6, 8,10, 9, 2)$
$ 6, 6, 3 $	$15$	$6$	$( 1, 3,11,13, 6, 8)( 2, 4,12,14, 7, 9)( 5,10,15)$
$ 6, 3, 3, 3 $	$10$	$6$	$( 1, 3,11,13, 6, 8)( 2, 7,12)( 4, 9,14)( 5,10,15)$
$ 5, 5, 5 $	$24$	$5$	$( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)$
$ 3, 3, 3, 2, 2, 2 $	$20$	$6$	$( 1, 4,13)( 2, 5)( 3, 6, 9)( 7,10)( 8,11,14)(12,15)$
$ 3, 3, 3, 3, 3 $	$1$	$3$	$( 1, 6,11)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$
$ 3, 3, 3, 3, 3 $	$1$	$3$	$( 1,11, 6)( 2,12, 7)( 3,13, 8)( 4,14, 9)( 5,15,10)$

Group invariants

Order:		$360=2^{3} \cdot 3^{2} \cdot 5$
Cyclic:		No
Abelian:		No
Solvable:		No
GAP id:		[360, 119]

Character table: Data not available.