Galois group: 15T12

• Label: 15T12
• Degree: $15$
• Order: $150$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $(C_5^2 : C_3):C_2$

Group invariants

Abstract group:		$(C_5^2 : C_3):C_2$
Order:		$150=2 \cdot 3 \cdot 5^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$15$
Transitive number $t$:		$12$
CHM label:		$[5^{2}:2]3$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$1$
Generators:		$(1,4)(2,8)(3,12)(6,9)(7,13)(11,14)$, $(1,13,10,7,4)(2,5,8,11,14)$, $(1,6,11)(2,7,12)(3,8,13)(4,9,14)(5,10,15)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$6$:	$C_6$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 5: None

Low degree siblings

15T12, 25T15, 30T35 x 2

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	Index	Representative
1A	$1^{15}$	$1$	$1$	$0$	$()$
2A	$2^{6},1^{3}$	$25$	$2$	$6$	$( 1,13)( 2, 8)( 4,10)( 6,15)( 9,12)(11,14)$
3A1	$3^{5}$	$25$	$3$	$10$	$( 1,12,14)( 2, 4,15)( 3, 5, 7)( 6, 8,10)( 9,11,13)$
3A-1	$3^{5}$	$25$	$3$	$10$	$( 1,14,12)( 2,15, 4)( 3, 7, 5)( 6,10, 8)( 9,13,11)$
5A1	$5^{3}$	$6$	$5$	$12$	$( 1, 7,13, 4,10)( 2,14,11, 8, 5)( 3,15,12, 9, 6)$
5A2	$5^{3}$	$6$	$5$	$12$	$( 1,13,10, 7, 4)( 2,11, 5,14, 8)( 3,12, 6,15, 9)$
5B1	$5^{2},1^{5}$	$6$	$5$	$8$	$( 1, 4, 7,10,13)( 2,14,11, 8, 5)$
5B2	$5^{2},1^{5}$	$6$	$5$	$8$	$( 1, 7,13, 4,10)( 2,11, 5,14, 8)$
6A1	$6^{2},3$	$25$	$6$	$12$	$( 1,11,12,13,14, 9)( 2, 6, 4, 8,15,10)( 3, 7, 5)$
6A-1	$6^{2},3$	$25$	$6$	$12$	$( 1, 9,14,13,12,11)( 2,10,15, 8, 4, 6)( 3, 5, 7)$

Malle's constant $a(G)$:     $1/6$

Character table

		1A	2A	3A1	3A-1	5A1	5A2	5B1	5B2	6A1	6A-1
	Size	1	25	25	25	6	6	6	6	25	25
	2 P	1A	1A	3A-1	3A1	5A2	5A1	5B2	5B1	3A1	3A-1
	3 P	1A	2A	1A	1A	5A2	5A1	5B2	5B1	2A	2A
	5 P	1A	2A	3A-1	3A1	1A	1A	1A	1A	6A-1	6A1
	Type
150.6.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
150.6.1b	R	$1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$
150.6.1c1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$
150.6.1c2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$
150.6.1d1	C	$1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	$1$	$1$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$
150.6.1d2	C	$1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	$1$	$1$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$
150.6.6a1	R	$6$	$0$	$0$	$0$	${\zeta }_5^{-2}-1+{\zeta }_5^2$	$-{\zeta }_5^{-2}-2-{\zeta }_5^2$	$-2{\zeta }_5^{-2}-2{\zeta }_5^2$	$-2{\zeta }_5^{-1}-2{\zeta }_5$	$0$	$0$
150.6.6a2	R	$6$	$0$	$0$	$0$	$-{\zeta }_5^{-2}-2-{\zeta }_5^2$	${\zeta }_5^{-2}-1+{\zeta }_5^2$	$-2{\zeta }_5^{-1}-2{\zeta }_5$	$-2{\zeta }_5^{-2}-2{\zeta }_5^2$	$0$	$0$
150.6.6b1	R	$6$	$0$	$0$	$0$	$-2{\zeta }_5^{-1}-2{\zeta }_5$	$-2{\zeta }_5^{-2}-2{\zeta }_5^2$	${\zeta }_5^{-2}-1+{\zeta }_5^2$	$-{\zeta }_5^{-2}-2-{\zeta }_5^2$	$0$	$0$
150.6.6b2	R	$6$	$0$	$0$	$0$	$-2{\zeta }_5^{-2}-2{\zeta }_5^2$	$-2{\zeta }_5^{-1}-2{\zeta }_5$	$-{\zeta }_5^{-2}-2-{\zeta }_5^2$	${\zeta }_5^{-2}-1+{\zeta }_5^2$	$0$	$0$

Regular extensions

Data not computed