Galois Group: 30T11

• Label: 30T11
• Order: \(60\)
• n: \(30\)
• Cyclic: No
• Abelian: No
• Solvable: Yes
• Primitive: No
• $p$-group: No
• Group: $C_5\times A_4$

Group action invariants

Degree $n$ :		$30$
Transitive number $t$ :		$11$
Group :		$C_5\times A_4$
Parity:		$1$
Primitive:		No
Nilpotency class:		$-1$ (not nilpotent)
Generators:		(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,20), (1,20,29,10,17,28,7,15,25,5,14,23,4,11,21)(2,19,30,9,18,27,8,16,26,6,13,24,3,12,22)
$|\Aut(F/K)|$:		$10$

Low degree resolvents

|G/N|	Galois groups for stem field(s)
3:	$C_3$
5:	$C_5$
12:	$A_4$
15:	$C_{15}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 5: $C_5$

Degree 6: $A_4$

Degree 10: None

Degree 15: $C_{15}$

Low degree siblings

20T14

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, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$3$	$2$	$(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)$
$ 10, 10, 5, 5 $	$3$	$10$	$( 1, 3, 5, 8,10, 2, 4, 6, 7, 9)(11,13,15,18,20,12,14,16,17,19)(21,23,25,28,29) (22,24,26,27,30)$
$ 5, 5, 5, 5, 5, 5 $	$1$	$5$	$( 1, 4, 5, 7,10)( 2, 3, 6, 8, 9)(11,14,15,17,20)(12,13,16,18,19) (21,23,25,28,29)(22,24,26,27,30)$
$ 5, 5, 5, 5, 5, 5 $	$1$	$5$	$( 1, 5,10, 4, 7)( 2, 6, 9, 3, 8)(11,15,20,14,17)(12,16,19,13,18) (21,25,29,23,28)(22,26,30,24,27)$
$ 10, 10, 5, 5 $	$3$	$10$	$( 1, 5,10, 4, 7)( 2, 6, 9, 3, 8)(11,16,20,13,17,12,15,19,14,18) (21,26,29,24,28,22,25,30,23,27)$
$ 5, 5, 5, 5, 5, 5 $	$1$	$5$	$( 1, 7, 4,10, 5)( 2, 8, 3, 9, 6)(11,17,14,20,15)(12,18,13,19,16) (21,28,23,29,25)(22,27,24,30,26)$
$ 10, 10, 5, 5 $	$3$	$10$	$( 1, 7, 4,10, 5)( 2, 8, 3, 9, 6)(11,18,14,19,15,12,17,13,20,16) (21,27,23,30,25,22,28,24,29,26)$
$ 10, 10, 5, 5 $	$3$	$10$	$( 1, 9, 7, 6, 4, 2,10, 8, 5, 3)(11,19,17,16,14,12,20,18,15,13)(21,29,28,25,23) (22,30,27,26,24)$
$ 5, 5, 5, 5, 5, 5 $	$1$	$5$	$( 1,10, 7, 5, 4)( 2, 9, 8, 6, 3)(11,20,17,15,14)(12,19,18,16,13) (21,29,28,25,23)(22,30,27,26,24)$
$ 15, 15 $	$4$	$15$	$( 1,11,23, 5,15,28,10,20,21, 4,14,25, 7,17,29)( 2,12,24, 6,16,27, 9,19,22, 3, 13,26, 8,18,30)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$4$	$3$	$( 1,13,27)( 2,14,28)( 3,15,29)( 4,16,30)( 5,18,22)( 6,17,21)( 7,19,24) ( 8,20,23)( 9,11,25)(10,12,26)$
$ 15, 15 $	$4$	$15$	$( 1,15,22, 7,11,27, 4,17,24,10,14,30, 5,20,26)( 2,16,21, 8,12,28, 3,18,23, 9, 13,29, 6,19,25)$
$ 15, 15 $	$4$	$15$	$( 1,17,25, 4,20,28, 5,11,29, 7,14,21,10,15,23)( 2,18,26, 3,19,27, 6,12,30, 8, 13,22, 9,16,24)$
$ 15, 15 $	$4$	$15$	$( 1,19,30,10,18,27, 7,16,26, 5,13,24, 4,12,22)( 2,20,29, 9,17,28, 8,15,25, 6, 14,23, 3,11,21)$
$ 15, 15 $	$4$	$15$	$( 1,21,12, 4,23,13, 5,25,16, 7,28,18,10,29,19)( 2,22,11, 3,24,14, 6,26,15, 8, 27,17, 9,30,20)$
$ 15, 15 $	$4$	$15$	$( 1,23,15,10,21,14, 7,29,11, 5,28,20, 4,25,17)( 2,24,16, 9,22,13, 8,30,12, 6, 27,19, 3,26,18)$
$ 15, 15 $	$4$	$15$	$( 1,25,20, 5,29,14,10,23,17, 4,28,11, 7,21,15)( 2,26,19, 6,30,13, 9,24,18, 3, 27,12, 8,22,16)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$4$	$3$	$( 1,27,13)( 2,28,14)( 3,29,15)( 4,30,16)( 5,22,18)( 6,21,17)( 7,24,19) ( 8,23,20)( 9,25,11)(10,26,12)$
$ 15, 15 $	$4$	$15$	$( 1,29,17, 7,25,14, 4,21,20,10,28,15, 5,23,11)( 2,30,18, 8,26,13, 3,22,19, 9, 27,16, 6,24,12)$

Group invariants

Order:		$60=2^{2} \cdot 3 \cdot 5$
Cyclic:		No
Abelian:		No
Solvable:		Yes
GAP id:		[60, 9]

Character table:

2 2 2 2 2 2 2 2 2 2 2 . . . . . . . . . 3 1 . . 1 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 1 1 1 1 1 1 1 1a 2a 10a 5a 5b 10b 5c 10c 10d 5d 15a 3a 15b 15c 15d 15e 15f 15g 3b 2P 1a 1a 5b 5b 5d 5d 5a 5a 5c 5c 15f 3b 15e 15g 15h 15a 15b 15d 3a 3P 1a 2a 10c 5c 5a 10a 5d 10d 10b 5b 5b 1a 5c 5a 5d 5a 5d 5b 1a 5P 1a 2a 2a 1a 1a 2a 1a 2a 2a 1a 3b 3b 3b 3b 3b 3a 3a 3a 3a 7P 1a 2a 10b 5b 5d 10d 5a 10a 10c 5c 15d 3a 15c 15a 15b 15g 15h 15f 3b 11P 1a 2a 10a 5a 5b 10b 5c 10c 10d 5d 15g 3b 15h 15e 15f 15c 15d 15a 3a 13P 1a 2a 10c 5c 5a 10a 5d 10d 10b 5b 15c 3a 15d 15b 15a 15h 15g 15e 3b X.1 1 1 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 E E E E E /E /E /E /E X.3 1 1 1 1 1 1 1 1 1 1 /E /E /E /E /E E E E E X.4 1 1 A A B B /B /B /A /A /A 1 A B /B B /B /A 1 X.5 1 1 B B /A /A A A /B /B /B 1 B /A A /A A /B 1 X.6 1 1 /B /B A A /A /A B B B 1 /B A /A A /A B 1 X.7 1 1 /A /A /B /B B B A A A 1 /A /B B /B B A 1 X.8 1 1 A A B B /B /B /A /A F E /G /H I /I H G /E X.9 1 1 A A B B /B /B /A /A G /E /F /I H /H I F E X.10 1 1 B B /A /A A A /B /B H /E /I G /F F /G I E X.11 1 1 B B /A /A A A /B /B I E /H F /G G /F H /E X.12 1 1 /B /B A A /A /A B B /I /E H /F G /G F /H E X.13 1 1 /B /B A A /A /A B B /H E I /G F /F G /I /E X.14 1 1 /A /A /B /B B B A A /G E F I /H H /I /F /E X.15 1 1 /A /A /B /B B B A A /F /E G H /I I /H /G E X.16 3 -1 -1 3 3 -1 3 -1 -1 3 . . . . . . . . . X.17 3 -1 -/B C /D -A D -/A -B /C . . . . . . . . . X.18 3 -1 -/A D C -/B /C -B -A /D . . . . . . . . . X.19 3 -1 -A /D /C -B C -/B -/A D . . . . . . . . . X.20 3 -1 -B /C D -/A /D -A -/B C . . . . . . . . . 2 . 3 1 5 1 15h 2P 15c 3P 5c 5P 3a 7P 15e 11P 15b 13P 15f X.1 1 X.2 /E X.3 E X.4 A X.5 B X.6 /B X.7 /A X.8 /F X.9 /G X.10 /H X.11 /I X.12 I X.13 H X.14 G X.15 F X.16 . X.17 . X.18 . X.19 . X.20 . A = E(5)^4 B = E(5)^3 C = 3*E(5)^2 D = 3*E(5) E = E(3)^2 = (-1-Sqrt(-3))/2 = -1-b3 F = E(15)^13 G = E(15)^8 H = E(15)^11 I = E(15)