Galois group: 20T37

• Label: 20T37
• Degree: $20$
• Order: $120$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $D_5\times A_4$

Group action invariants

Degree $n$:		$20$
Transitive number $t$:		$37$
Group:		$D_5\times A_4$
Parity:		$1$
Primitive:		no
Nilpotency class:		$-1$ (not nilpotent)
$|\Aut(F/K)|$:		$1$
Generators:		(1,16,2,13,4,14)(3,15)(5,12,6,9,8,10)(7,11)(17,20,18), (1,8,9,16,17,4,5,12,13,20)(2,7,10,15,18,3,6,11,14,19)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$6$:	$C_6$
$10$:	$D_{5}$
$12$:	$A_4$
$24$:	$A_4\times C_2$
$30$:	$D_5\times C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: $A_4$

Degree 5: $D_{5}$

Degree 10: None

Low degree siblings

30T20, 30T28, 40T65

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, 1, 1, 1 $	$1$	$1$	$()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $	$5$	$2$	$( 5,17)( 6,18)( 7,19)( 8,20)( 9,13)(10,14)(11,15)(12,16)$
$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $	$4$	$3$	$( 2, 3, 4)( 6, 7, 8)(10,11,12)(14,15,16)(18,19,20)$
$ 6, 6, 3, 2, 2, 1 $	$20$	$6$	$( 2, 3, 4)( 5,17)( 6,19, 8,18, 7,20)( 9,13)(10,15,12,14,11,16)$
$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1 $	$4$	$3$	$( 2, 4, 3)( 6, 8, 7)(10,12,11)(14,16,15)(18,20,19)$
$ 6, 6, 3, 2, 2, 1 $	$20$	$6$	$( 2, 4, 3)( 5,17)( 6,20, 7,18, 8,19)( 9,13)(10,16,11,14,12,15)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$3$	$2$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$15$	$2$	$( 1, 2)( 3, 4)( 5,18)( 6,17)( 7,20)( 8,19)( 9,14)(10,13)(11,16)(12,15)$
$ 5, 5, 5, 5 $	$2$	$5$	$( 1, 5, 9,13,17)( 2, 6,10,14,18)( 3, 7,11,15,19)( 4, 8,12,16,20)$
$ 15, 5 $	$8$	$15$	$( 1, 5, 9,13,17)( 2, 7,12,14,19, 4, 6,11,16,18, 3, 8,10,15,20)$
$ 15, 5 $	$8$	$15$	$( 1, 5, 9,13,17)( 2, 8,11,14,20, 3, 6,12,15,18, 4, 7,10,16,19)$
$ 10, 10 $	$6$	$10$	$( 1, 6, 9,14,17, 2, 5,10,13,18)( 3, 8,11,16,19, 4, 7,12,15,20)$
$ 5, 5, 5, 5 $	$2$	$5$	$( 1, 9,17, 5,13)( 2,10,18, 6,14)( 3,11,19, 7,15)( 4,12,20, 8,16)$
$ 15, 5 $	$8$	$15$	$( 1, 9,17, 5,13)( 2,11,20, 6,15, 4,10,19, 8,14, 3,12,18, 7,16)$
$ 15, 5 $	$8$	$15$	$( 1, 9,17, 5,13)( 2,12,19, 6,16, 3,10,20, 7,14, 4,11,18, 8,15)$
$ 10, 10 $	$6$	$10$	$( 1,10,17, 6,13, 2, 9,18, 5,14)( 3,12,19, 8,15, 4,11,20, 7,16)$

Group invariants

Order:		$120=2^{3} \cdot 3 \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		yes
GAP id:		[120, 39]

Character table:

2 3 3 1 1 1 1 3 3 2 . . 2 2 . . 2 3 1 1 1 1 1 1 . . 1 1 1 . 1 1 1 . 5 1 . 1 . 1 . 1 . 1 1 1 1 1 1 1 1 1a 2a 3a 6a 3b 6b 2b 2c 5a 15a 15b 10a 5b 15c 15d 10b 2P 1a 1a 3b 3b 3a 3a 1a 1a 5b 15d 15c 5b 5a 15b 15a 5a 3P 1a 2a 1a 2a 1a 2a 2b 2c 5b 5b 5b 10b 5a 5a 5a 10a 5P 1a 2a 3b 6b 3a 6a 2b 2c 1a 3b 3a 2b 1a 3b 3a 2b 7P 1a 2a 3a 6a 3b 6b 2b 2c 5b 15c 15d 10b 5a 15a 15b 10a 11P 1a 2a 3b 6b 3a 6a 2b 2c 5a 15b 15a 10a 5b 15d 15c 10b 13P 1a 2a 3a 6a 3b 6b 2b 2c 5b 15c 15d 10b 5a 15a 15b 10a X.1 1 1 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 1 1 1 X.3 1 -1 A -A /A -/A 1 -1 1 A /A 1 1 A /A 1 X.4 1 -1 /A -/A A -A 1 -1 1 /A A 1 1 /A A 1 X.5 1 1 A A /A /A 1 1 1 A /A 1 1 A /A 1 X.6 1 1 /A /A A A 1 1 1 /A A 1 1 /A A 1 X.7 2 . 2 . 2 . 2 . C C C C *C *C *C *C X.8 2 . 2 . 2 . 2 . *C *C *C *C C C C C X.9 2 . B . /B . 2 . C E /E C *C F /F *C X.10 2 . /B . B . 2 . C /E E C *C /F F *C X.11 2 . B . /B . 2 . *C F /F *C C E /E C X.12 2 . /B . B . 2 . *C /F F *C C /E E C X.13 3 -3 . . . . -1 1 3 . . -1 3 . . -1 X.14 3 3 . . . . -1 -1 3 . . -1 3 . . -1 X.15 6 . . . . . -2 . D . . -C *D . . -*C X.16 6 . . . . . -2 . *D . . -*C D . . -C A = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 B = 2*E(3)^2 = -1-Sqrt(-3) = -1-i3 C = E(5)+E(5)^4 = (-1+Sqrt(5))/2 = b5 D = 3*E(5)+3*E(5)^4 = (-3+3*Sqrt(5))/2 = 3b5 E = E(15)^7+E(15)^13 F = E(15)+E(15)^4