Galois Group: 15T42

• Label: 15T42
• Order: \(1620\)
• n: \(15\)
• Cyclic: No
• Abelian: No
• Solvable: Yes
• Primitive: No
• $p$-group: No

Group action invariants

Degree $n$ :		$15$
Transitive number $t$ :		$42$
CHM label :		$1/2[3^{4}:2]F(5)$
Parity:		$1$
Primitive:		No
Nilpotency class:		$-1$ (not nilpotent)
Generators:		(1,2,4,8)(3,6,12,9)(5,10)(7,14,13,11), (1,6,11)(4,14,9), (1,4,7,10,13)(2,5,8,11,14)(3,6,9,12,15)
$|\Aut(F/K)|$:		$1$

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: $F_5$

Low degree siblings

15T41, 30T295, 30T296, 30T298, 30T299, 30T302, 45T204, 45T210

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 $	$1$	$1$	$()$
$ 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$20$	$3$	$( 1,11, 6)( 4, 9,14)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $	$10$	$3$	$( 1, 6,11)( 2, 7,12)( 4, 9,14)$
$ 3, 3, 3, 1, 1, 1, 1, 1, 1 $	$10$	$3$	$( 1,11, 6)( 2,12, 7)( 4,14, 9)$
$ 3, 3, 3, 3, 1, 1, 1 $	$5$	$3$	$( 1,11, 6)( 2, 7,12)( 3, 8,13)( 4,14, 9)$
$ 3, 3, 3, 3, 1, 1, 1 $	$20$	$3$	$( 1,11, 6)( 2,12, 7)( 3, 8,13)( 4, 9,14)$
$ 3, 3, 3, 3, 1, 1, 1 $	$5$	$3$	$( 1, 6,11)( 2,12, 7)( 3,13, 8)( 4, 9,14)$
$ 3, 3, 3, 3, 3 $	$10$	$3$	$( 1,11, 6)( 2, 7,12)( 3, 8,13)( 4, 9,14)( 5,10,15)$
$ 5, 5, 5 $	$324$	$5$	$( 1, 4,12,10, 3)( 2,15, 8, 6, 9)( 5,13,11,14, 7)$
$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $	$45$	$2$	$( 2,15)( 3, 4)( 5, 7)( 8, 9)(10,12)(13,14)$
$ 6, 3, 2, 2, 2 $	$90$	$6$	$( 1,11, 6)( 2,15)( 3, 9, 8,14,13, 4)( 5, 7)(10,12)$
$ 6, 3, 2, 2, 2 $	$90$	$6$	$( 1, 6,11)( 2,15)( 3,14,13, 9, 8, 4)( 5, 7)(10,12)$
$ 6, 6, 1, 1, 1 $	$45$	$6$	$( 2,15, 7, 5,12,10)( 3,14,13, 9, 8, 4)$
$ 6, 6, 3 $	$90$	$6$	$( 1, 6,11)( 2,15, 7, 5,12,10)( 3, 9, 8,14,13, 4)$
$ 6, 6, 1, 1, 1 $	$45$	$6$	$( 2,15,12,10, 7, 5)( 3, 9, 8,14,13, 4)$
$ 12, 2, 1 $	$135$	$12$	$( 2, 3,15,14, 7,13, 5, 9,12, 8,10, 4)( 6,11)$
$ 4, 4, 4, 2, 1 $	$135$	$4$	$( 1,11)( 2, 3,15, 4)( 5,14, 7,13)( 8,10, 9,12)$
$ 12, 2, 1 $	$135$	$12$	$( 1, 6)( 2, 3,15, 9,12, 8,10,14, 7,13, 5, 4)$
$ 12, 2, 1 $	$135$	$12$	$( 2, 4,15,13, 7,14, 5, 8,12, 9,10, 3)( 6,11)$
$ 4, 4, 4, 2, 1 $	$135$	$4$	$( 1,11)( 2, 9,10, 3)( 4,15,13, 7)( 5, 8,12,14)$
$ 12, 2, 1 $	$135$	$12$	$( 1, 6)( 2,14, 5, 8,12, 4,15,13, 7, 9,10, 3)$

Group invariants

Order:		$1620=2^{2} \cdot 3^{4} \cdot 5$
Cyclic:		No
Abelian:		No
Solvable:		Yes
GAP id:		[1620, 421]

Character table: Data not available.