Galois group: 30T46

• Label: 30T46
• Degree: $30$
• Order: $180$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $C_3^2:F_5$

Group action invariants

Degree $n$:		$30$
Transitive number $t$:		$46$
Group:		$C_3^2:F_5$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$1$
Generators:		(1,12,5,14,8,2,10,6,15,9,3,11,4,13,7)(16,25,20,30,23,18,27,19,29,22,17,26,21,28,24), (1,20,6,28)(2,21,5,30)(3,19,4,29)(7,22,15,27)(8,23,14,26)(9,24,13,25)(10,18,12,17)(11,16)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$
$20$:	$F_5$
$36$:	$C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 3: None

Degree 5: $F_5$

Degree 6: $C_3^2:C_4$

Degree 10: $F_5$

Degree 15: None

Low degree siblings

30T46, 45T27

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Label	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 $	$1$	$1$	$()$
	$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$4$	$3$	$(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $	$45$	$2$	$( 2, 3)( 4,14)( 5,13)( 6,15)( 7,12)( 8,11)( 9,10)(16,28)(17,30)(18,29)(19,27) (20,26)(21,25)(22,23)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$4$	$3$	$( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)$
	$ 15, 5, 5, 5 $	$4$	$15$	$( 1, 4, 9,10,14)( 2, 5, 7,11,15)( 3, 6, 8,12,13)(16,19,24,27,28,18,21,23,26, 30,17,20,22,25,29)$
	$ 15, 5, 5, 5 $	$4$	$15$	$( 1, 4, 9,10,14)( 2, 5, 7,11,15)( 3, 6, 8,12,13)(16,20,23,27,29,17,21,24,25, 30,18,19,22,26,28)$
	$ 5, 5, 5, 5, 5, 5 $	$4$	$5$	$( 1, 4, 9,10,14)( 2, 5, 7,11,15)( 3, 6, 8,12,13)(16,21,22,27,30) (17,19,23,25,28)(18,20,24,26,29)$
	$ 15, 15 $	$4$	$15$	$( 1, 5, 8,10,15, 3, 4, 7,12,14, 2, 6, 9,11,13)(16,19,24,27,28,18,21,23,26,30, 17,20,22,25,29)$
	$ 15, 15 $	$4$	$15$	$( 1, 5, 8,10,15, 3, 4, 7,12,14, 2, 6, 9,11,13)(16,20,23,27,29,17,21,24,25,30, 18,19,22,26,28)$
	$ 15, 5, 5, 5 $	$4$	$15$	$( 1, 5, 8,10,15, 3, 4, 7,12,14, 2, 6, 9,11,13)(16,21,22,27,30)(17,19,23,25,28) (18,20,24,26,29)$
	$ 15, 15 $	$4$	$15$	$( 1, 6, 7,10,13, 2, 4, 8,11,14, 3, 5, 9,12,15)(16,19,24,27,28,18,21,23,26,30, 17,20,22,25,29)$
	$ 15, 15 $	$4$	$15$	$( 1, 6, 7,10,13, 2, 4, 8,11,14, 3, 5, 9,12,15)(16,20,23,27,29,17,21,24,25,30, 18,19,22,26,28)$
	$ 15, 5, 5, 5 $	$4$	$15$	$( 1, 6, 7,10,13, 2, 4, 8,11,14, 3, 5, 9,12,15)(16,21,22,27,30)(17,19,23,25,28) (18,20,24,26,29)$
	$ 4, 4, 4, 4, 4, 4, 4, 2 $	$45$	$4$	$( 1,16, 7,19)( 2,17, 9,21)( 3,18, 8,20)( 4,27, 5,25)( 6,26)(10,30,15,23) (11,28,14,22)(12,29,13,24)$
	$ 4, 4, 4, 4, 4, 4, 4, 2 $	$45$	$4$	$( 1,16,13,25)( 2,18,15,26)( 3,17,14,27)( 4,22,12,19)( 5,24,11,20)( 6,23,10,21) ( 7,29)( 8,28, 9,30)$

Group invariants

Order:		$180=2^{2} \cdot 3^{2} \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		180.25
Character table:

		1A	2A	3A	3B	4A1	4A-1	5A	15A1	15A2	15A4	15A7	15B1	15B2	15B4	15B7
	Size	1	45	4	4	45	45	4	4	4	4	4	4	4	4	4
	2 P	1A	1A	3A	3B	2A	2A	5A	15A2	15A4	15A1	15B4	15A7	15B1	15B7	15B2
	3 P	1A	2A	1A	1A	4A-1	4A1	5A	5A	5A	5A	5A	5A	5A	5A	5A
	5 P	1A	2A	3A	3B	4A1	4A-1	1A	3A	3A	3A	3B	3A	3B	3B	3B
	Type
180.25.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
180.25.1b	R	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
180.25.1c1	C	$1$	$-1$	$1$	$1$	$-i$	$i$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
180.25.1c2	C	$1$	$-1$	$1$	$1$	$i$	$-i$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
180.25.4a	R	$4$	$0$	$4$	$4$	$0$	$0$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$
180.25.4b	R	$4$	$0$	$-2$	$1$	$0$	$0$	$4$	$-2$	$-2$	$-2$	$-2$	$1$	$1$	$1$	$1$
180.25.4c	R	$4$	$0$	$1$	$-2$	$0$	$0$	$4$	$1$	$1$	$1$	$1$	$-2$	$-2$	$-2$	$-2$
180.25.4d1	R	$4$	$0$	$-2$	$1$	$0$	$0$	$-1$	$-1-{\zeta }_{15}^3+2{\zeta }_{15}^4-{\zeta }_{15}^5-{\zeta }_{15}^6+2{\zeta }_{15}^7$	$2{\zeta }_{15}-{\zeta }_{15}^2-{\zeta }_{15}^5+{\zeta }_{15}^6+{\zeta }_{15}^7$	$2+{\zeta }_{15}^3-2{\zeta }_{15}^4+{\zeta }_{15}^5+{\zeta }_{15}^6-2{\zeta }_{15}^7$	$1-2{\zeta }_{15}+{\zeta }_{15}^2+{\zeta }_{15}^5-{\zeta }_{15}^6-{\zeta }_{15}^7$	$-2+{\zeta }_{15}+{\zeta }_{15}^2-2{\zeta }_{15}^3+{\zeta }_{15}^4-{\zeta }_{15}^5+3{\zeta }_{15}^7$	$1+{\zeta }_{15}-2{\zeta }_{15}^2+2{\zeta }_{15}^3-{\zeta }_{15}^4+{\zeta }_{15}^6-2{\zeta }_{15}^7$	$-{\zeta }_{15}+2{\zeta }_{15}^2-{\zeta }_{15}^3-{\zeta }_{15}^4+{\zeta }_{15}^5$	$-{\zeta }_{15}-{\zeta }_{15}^2+{\zeta }_{15}^3+{\zeta }_{15}^4-{\zeta }_{15}^6-{\zeta }_{15}^7$
180.25.4d2	R	$4$	$0$	$-2$	$1$	$0$	$0$	$-1$	$2{\zeta }_{15}-{\zeta }_{15}^2-{\zeta }_{15}^5+{\zeta }_{15}^6+{\zeta }_{15}^7$	$2+{\zeta }_{15}^3-2{\zeta }_{15}^4+{\zeta }_{15}^5+{\zeta }_{15}^6-2{\zeta }_{15}^7$	$1-2{\zeta }_{15}+{\zeta }_{15}^2+{\zeta }_{15}^5-{\zeta }_{15}^6-{\zeta }_{15}^7$	$-1-{\zeta }_{15}^3+2{\zeta }_{15}^4-{\zeta }_{15}^5-{\zeta }_{15}^6+2{\zeta }_{15}^7$	$1+{\zeta }_{15}-2{\zeta }_{15}^2+2{\zeta }_{15}^3-{\zeta }_{15}^4+{\zeta }_{15}^6-2{\zeta }_{15}^7$	$-{\zeta }_{15}+2{\zeta }_{15}^2-{\zeta }_{15}^3-{\zeta }_{15}^4+{\zeta }_{15}^5$	$-{\zeta }_{15}-{\zeta }_{15}^2+{\zeta }_{15}^3+{\zeta }_{15}^4-{\zeta }_{15}^6-{\zeta }_{15}^7$	$-2+{\zeta }_{15}+{\zeta }_{15}^2-2{\zeta }_{15}^3+{\zeta }_{15}^4-{\zeta }_{15}^5+3{\zeta }_{15}^7$
180.25.4d3	R	$4$	$0$	$-2$	$1$	$0$	$0$	$-1$	$1-2{\zeta }_{15}+{\zeta }_{15}^2+{\zeta }_{15}^5-{\zeta }_{15}^6-{\zeta }_{15}^7$	$-1-{\zeta }_{15}^3+2{\zeta }_{15}^4-{\zeta }_{15}^5-{\zeta }_{15}^6+2{\zeta }_{15}^7$	$2{\zeta }_{15}-{\zeta }_{15}^2-{\zeta }_{15}^5+{\zeta }_{15}^6+{\zeta }_{15}^7$	$2+{\zeta }_{15}^3-2{\zeta }_{15}^4+{\zeta }_{15}^5+{\zeta }_{15}^6-2{\zeta }_{15}^7$	$-{\zeta }_{15}-{\zeta }_{15}^2+{\zeta }_{15}^3+{\zeta }_{15}^4-{\zeta }_{15}^6-{\zeta }_{15}^7$	$-2+{\zeta }_{15}+{\zeta }_{15}^2-2{\zeta }_{15}^3+{\zeta }_{15}^4-{\zeta }_{15}^5+3{\zeta }_{15}^7$	$1+{\zeta }_{15}-2{\zeta }_{15}^2+2{\zeta }_{15}^3-{\zeta }_{15}^4+{\zeta }_{15}^6-2{\zeta }_{15}^7$	$-{\zeta }_{15}+2{\zeta }_{15}^2-{\zeta }_{15}^3-{\zeta }_{15}^4+{\zeta }_{15}^5$
180.25.4d4	R	$4$	$0$	$-2$	$1$	$0$	$0$	$-1$	$2+{\zeta }_{15}^3-2{\zeta }_{15}^4+{\zeta }_{15}^5+{\zeta }_{15}^6-2{\zeta }_{15}^7$	$1-2{\zeta }_{15}+{\zeta }_{15}^2+{\zeta }_{15}^5-{\zeta }_{15}^6-{\zeta }_{15}^7$	$-1-{\zeta }_{15}^3+2{\zeta }_{15}^4-{\zeta }_{15}^5-{\zeta }_{15}^6+2{\zeta }_{15}^7$	$2{\zeta }_{15}-{\zeta }_{15}^2-{\zeta }_{15}^5+{\zeta }_{15}^6+{\zeta }_{15}^7$	$-{\zeta }_{15}+2{\zeta }_{15}^2-{\zeta }_{15}^3-{\zeta }_{15}^4+{\zeta }_{15}^5$	$-{\zeta }_{15}-{\zeta }_{15}^2+{\zeta }_{15}^3+{\zeta }_{15}^4-{\zeta }_{15}^6-{\zeta }_{15}^7$	$-2+{\zeta }_{15}+{\zeta }_{15}^2-2{\zeta }_{15}^3+{\zeta }_{15}^4-{\zeta }_{15}^5+3{\zeta }_{15}^7$	$1+{\zeta }_{15}-2{\zeta }_{15}^2+2{\zeta }_{15}^3-{\zeta }_{15}^4+{\zeta }_{15}^6-2{\zeta }_{15}^7$
180.25.4e1	R	$4$	$0$	$1$	$-2$	$0$	$0$	$-1$	$-2+{\zeta }_{15}+{\zeta }_{15}^2-2{\zeta }_{15}^3+{\zeta }_{15}^4-{\zeta }_{15}^5+3{\zeta }_{15}^7$	$1+{\zeta }_{15}-2{\zeta }_{15}^2+2{\zeta }_{15}^3-{\zeta }_{15}^4+{\zeta }_{15}^6-2{\zeta }_{15}^7$	$-{\zeta }_{15}+2{\zeta }_{15}^2-{\zeta }_{15}^3-{\zeta }_{15}^4+{\zeta }_{15}^5$	$-{\zeta }_{15}-{\zeta }_{15}^2+{\zeta }_{15}^3+{\zeta }_{15}^4-{\zeta }_{15}^6-{\zeta }_{15}^7$	$1-2{\zeta }_{15}+{\zeta }_{15}^2+{\zeta }_{15}^5-{\zeta }_{15}^6-{\zeta }_{15}^7$	$-1-{\zeta }_{15}^3+2{\zeta }_{15}^4-{\zeta }_{15}^5-{\zeta }_{15}^6+2{\zeta }_{15}^7$	$2{\zeta }_{15}-{\zeta }_{15}^2-{\zeta }_{15}^5+{\zeta }_{15}^6+{\zeta }_{15}^7$	$2+{\zeta }_{15}^3-2{\zeta }_{15}^4+{\zeta }_{15}^5+{\zeta }_{15}^6-2{\zeta }_{15}^7$
180.25.4e2	R	$4$	$0$	$1$	$-2$	$0$	$0$	$-1$	$-{\zeta }_{15}+2{\zeta }_{15}^2-{\zeta }_{15}^3-{\zeta }_{15}^4+{\zeta }_{15}^5$	$-{\zeta }_{15}-{\zeta }_{15}^2+{\zeta }_{15}^3+{\zeta }_{15}^4-{\zeta }_{15}^6-{\zeta }_{15}^7$	$-2+{\zeta }_{15}+{\zeta }_{15}^2-2{\zeta }_{15}^3+{\zeta }_{15}^4-{\zeta }_{15}^5+3{\zeta }_{15}^7$	$1+{\zeta }_{15}-2{\zeta }_{15}^2+2{\zeta }_{15}^3-{\zeta }_{15}^4+{\zeta }_{15}^6-2{\zeta }_{15}^7$	$2{\zeta }_{15}-{\zeta }_{15}^2-{\zeta }_{15}^5+{\zeta }_{15}^6+{\zeta }_{15}^7$	$2+{\zeta }_{15}^3-2{\zeta }_{15}^4+{\zeta }_{15}^5+{\zeta }_{15}^6-2{\zeta }_{15}^7$	$1-2{\zeta }_{15}+{\zeta }_{15}^2+{\zeta }_{15}^5-{\zeta }_{15}^6-{\zeta }_{15}^7$	$-1-{\zeta }_{15}^3+2{\zeta }_{15}^4-{\zeta }_{15}^5-{\zeta }_{15}^6+2{\zeta }_{15}^7$
180.25.4e3	R	$4$	$0$	$1$	$-2$	$0$	$0$	$-1$	$-{\zeta }_{15}-{\zeta }_{15}^2+{\zeta }_{15}^3+{\zeta }_{15}^4-{\zeta }_{15}^6-{\zeta }_{15}^7$	$-2+{\zeta }_{15}+{\zeta }_{15}^2-2{\zeta }_{15}^3+{\zeta }_{15}^4-{\zeta }_{15}^5+3{\zeta }_{15}^7$	$1+{\zeta }_{15}-2{\zeta }_{15}^2+2{\zeta }_{15}^3-{\zeta }_{15}^4+{\zeta }_{15}^6-2{\zeta }_{15}^7$	$-{\zeta }_{15}+2{\zeta }_{15}^2-{\zeta }_{15}^3-{\zeta }_{15}^4+{\zeta }_{15}^5$	$2+{\zeta }_{15}^3-2{\zeta }_{15}^4+{\zeta }_{15}^5+{\zeta }_{15}^6-2{\zeta }_{15}^7$	$1-2{\zeta }_{15}+{\zeta }_{15}^2+{\zeta }_{15}^5-{\zeta }_{15}^6-{\zeta }_{15}^7$	$-1-{\zeta }_{15}^3+2{\zeta }_{15}^4-{\zeta }_{15}^5-{\zeta }_{15}^6+2{\zeta }_{15}^7$	$2{\zeta }_{15}-{\zeta }_{15}^2-{\zeta }_{15}^5+{\zeta }_{15}^6+{\zeta }_{15}^7$
180.25.4e4	R	$4$	$0$	$1$	$-2$	$0$	$0$	$-1$	$1+{\zeta }_{15}-2{\zeta }_{15}^2+2{\zeta }_{15}^3-{\zeta }_{15}^4+{\zeta }_{15}^6-2{\zeta }_{15}^7$	$-{\zeta }_{15}+2{\zeta }_{15}^2-{\zeta }_{15}^3-{\zeta }_{15}^4+{\zeta }_{15}^5$	$-{\zeta }_{15}-{\zeta }_{15}^2+{\zeta }_{15}^3+{\zeta }_{15}^4-{\zeta }_{15}^6-{\zeta }_{15}^7$	$-2+{\zeta }_{15}+{\zeta }_{15}^2-2{\zeta }_{15}^3+{\zeta }_{15}^4-{\zeta }_{15}^5+3{\zeta }_{15}^7$	$-1-{\zeta }_{15}^3+2{\zeta }_{15}^4-{\zeta }_{15}^5-{\zeta }_{15}^6+2{\zeta }_{15}^7$	$2{\zeta }_{15}-{\zeta }_{15}^2-{\zeta }_{15}^5+{\zeta }_{15}^6+{\zeta }_{15}^7$	$2+{\zeta }_{15}^3-2{\zeta }_{15}^4+{\zeta }_{15}^5+{\zeta }_{15}^6-2{\zeta }_{15}^7$	$1-2{\zeta }_{15}+{\zeta }_{15}^2+{\zeta }_{15}^5-{\zeta }_{15}^6-{\zeta }_{15}^7$