Galois group: 15T15

• Label: 15T15
• Degree: $15$
• Order: $180$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: no
• $p$-group: no
• Group: $\GL(2,4)$

Group action invariants

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

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$3$:	$C_3$
$60$:	$A_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 5: $A_5$

Low degree siblings

15T15, 15T16, 18T90, 30T45, 36T176, 45T16

Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

Label	Cycle Type	Size	Order	Representative
	$ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 3, 3, 3, 3, 1, 1, 1 $	$20$	$3$	$( 2, 3,15)( 4, 7, 5)( 9,11,10)(12,13,14)$
	$ 2, 2, 2, 2, 2, 2, 1, 1, 1 $	$15$	$2$	$( 2, 5)( 3,13)( 4,10)( 7,14)( 9,15)(11,12)$
	$ 3, 3, 3, 3, 1, 1, 1 $	$20$	$3$	$( 2, 7,13)( 3,12, 9)( 4,11,14)( 5,15,10)$
	$ 15 $	$12$	$15$	$( 1, 2, 5, 3,12, 8, 9,15,11,14, 6, 4,10, 7,13)$
	$ 6, 6, 3 $	$15$	$6$	$( 1, 2, 6, 4, 8, 9)( 3,14, 7,12,11,13)( 5,15,10)$
	$ 5, 5, 5 $	$12$	$5$	$( 1, 2, 7,14, 5)( 3,13,15, 8, 9)( 4,11,12,10, 6)$
	$ 6, 6, 3 $	$15$	$6$	$( 1, 2, 8, 9, 6, 4)( 3, 7,11)( 5,14,15,13,10,12)$
	$ 5, 5, 5 $	$12$	$5$	$( 1, 2,10,13,11)( 3, 6, 4,15,14)( 5,12, 7, 8, 9)$
	$ 15 $	$12$	$15$	$( 1, 2,11,15,12, 6, 4, 3, 5,13, 8, 9, 7,10,14)$
	$ 3, 3, 3, 3, 3 $	$20$	$3$	$( 1, 2,12)( 3,11, 7)( 4,13, 6)( 5,10,15)( 8, 9,14)$
	$ 15 $	$12$	$15$	$( 1, 2,13, 5,11, 6, 4,14,10, 3, 8, 9,12,15, 7)$
	$ 15 $	$12$	$15$	$( 1, 2,14,11, 5, 8, 9,13, 7,15, 6, 4,12, 3,10)$
	$ 3, 3, 3, 3, 3 $	$1$	$3$	$( 1, 6, 8)( 2, 4, 9)( 3, 7,11)( 5,10,15)(12,13,14)$
	$ 3, 3, 3, 3, 3 $	$1$	$3$	$( 1, 8, 6)( 2, 9, 4)( 3,11, 7)( 5,15,10)(12,14,13)$

Group invariants

Order:		$180=2^{2} \cdot 3^{2} \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent
Label:		180.19
Character table:

		1A	2A	3A1	3A-1	3B	3C1	3C-1	5A1	5A2	6A1	6A-1	15A1	15A-1	15A2	15A-2
	Size	1	15	1	1	20	20	20	12	12	15	15	12	12	12	12
	2 P	1A	1A	3A-1	3A1	3C1	3B	3C-1	5A2	5A1	3A1	3A-1	15A-1	15A2	15A1	15A-2
	3 P	1A	2A	1A	1A	1A	1A	1A	5A2	5A1	2A	2A	5A2	5A1	5A2	5A1
	5 P	1A	2A	3A-1	3A1	3C1	3B	3C-1	1A	1A	6A-1	6A1	3A1	3A1	3A-1	3A-1
	Type
180.19.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
180.19.1b1	C	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
180.19.1b2	C	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
180.19.3a1	R	$3$	$-1$	$3$	$3$	$0$	$0$	$0$	$-{\zeta }_5^{-1}-{\zeta }_5$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$-1$	$-1$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$-{\zeta }_5^{-1}-{\zeta }_5$	$-{\zeta }_5^{-1}-{\zeta }_5$
180.19.3a2	R	$3$	$-1$	$3$	$3$	$0$	$0$	$0$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$-{\zeta }_5^{-1}-{\zeta }_5$	$-1$	$-1$	$-{\zeta }_5^{-1}-{\zeta }_5$	$-{\zeta }_5^{-1}-{\zeta }_5$	$-{\zeta }_5^{-2}-{\zeta }_5^2$	$-{\zeta }_5^{-2}-{\zeta }_5^2$
180.19.3b1	C	$3$	$-1$	$3{\zeta }_{15}^{-5}$	$3{\zeta }_{15}^5$	$0$	$0$	$0$	$-{\zeta }_{15}^{-3}-{\zeta }_{15}^3$	$-{\zeta }_{15}^{-6}-{\zeta }_{15}^6$	$-{\zeta }_{15}^5$	$-{\zeta }_{15}^{-5}$	$-1+{\zeta }_{15}+{\zeta }_{15}^2-{\zeta }_{15}^3+{\zeta }_{15}^4+{\zeta }_{15}^7$	$-{\zeta }_{15}-{\zeta }_{15}^4$	$-1+{\zeta }_{15}+{\zeta }_{15}^4-{\zeta }_{15}^5$	$1-{\zeta }_{15}-{\zeta }_{15}^2+{\zeta }_{15}^3-{\zeta }_{15}^4+{\zeta }_{15}^5-{\zeta }_{15}^7$
180.19.3b2	C	$3$	$-1$	$3{\zeta }_{15}^5$	$3{\zeta }_{15}^{-5}$	$0$	$0$	$0$	$-{\zeta }_{15}^{-3}-{\zeta }_{15}^3$	$-{\zeta }_{15}^{-6}-{\zeta }_{15}^6$	$-{\zeta }_{15}^{-5}$	$-{\zeta }_{15}^5$	$-{\zeta }_{15}-{\zeta }_{15}^4$	$-1+{\zeta }_{15}+{\zeta }_{15}^2-{\zeta }_{15}^3+{\zeta }_{15}^4+{\zeta }_{15}^7$	$1-{\zeta }_{15}-{\zeta }_{15}^2+{\zeta }_{15}^3-{\zeta }_{15}^4+{\zeta }_{15}^5-{\zeta }_{15}^7$	$-1+{\zeta }_{15}+{\zeta }_{15}^4-{\zeta }_{15}^5$
180.19.3b3	C	$3$	$-1$	$3{\zeta }_{15}^{-5}$	$3{\zeta }_{15}^5$	$0$	$0$	$0$	$-{\zeta }_{15}^{-6}-{\zeta }_{15}^6$	$-{\zeta }_{15}^{-3}-{\zeta }_{15}^3$	$-{\zeta }_{15}^5$	$-{\zeta }_{15}^{-5}$	$1-{\zeta }_{15}-{\zeta }_{15}^2+{\zeta }_{15}^3-{\zeta }_{15}^4+{\zeta }_{15}^5-{\zeta }_{15}^7$	$-1+{\zeta }_{15}+{\zeta }_{15}^4-{\zeta }_{15}^5$	$-{\zeta }_{15}-{\zeta }_{15}^4$	$-1+{\zeta }_{15}+{\zeta }_{15}^2-{\zeta }_{15}^3+{\zeta }_{15}^4+{\zeta }_{15}^7$
180.19.3b4	C	$3$	$-1$	$3{\zeta }_{15}^5$	$3{\zeta }_{15}^{-5}$	$0$	$0$	$0$	$-{\zeta }_{15}^{-6}-{\zeta }_{15}^6$	$-{\zeta }_{15}^{-3}-{\zeta }_{15}^3$	$-{\zeta }_{15}^{-5}$	$-{\zeta }_{15}^5$	$-1+{\zeta }_{15}+{\zeta }_{15}^4-{\zeta }_{15}^5$	$1-{\zeta }_{15}-{\zeta }_{15}^2+{\zeta }_{15}^3-{\zeta }_{15}^4+{\zeta }_{15}^5-{\zeta }_{15}^7$	$-1+{\zeta }_{15}+{\zeta }_{15}^2-{\zeta }_{15}^3+{\zeta }_{15}^4+{\zeta }_{15}^7$	$-{\zeta }_{15}-{\zeta }_{15}^4$
180.19.4a	R	$4$	$0$	$4$	$4$	$1$	$1$	$1$	$-1$	$-1$	$0$	$0$	$-1$	$-1$	$-1$	$-1$
180.19.4b1	C	$4$	$0$	$4{\zeta }_3^{-1}$	$4{\zeta }_3$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-1$	$-1$	$0$	$0$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$
180.19.4b2	C	$4$	$0$	$4{\zeta }_3$	$4{\zeta }_3^{-1}$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-1$	$-1$	$0$	$0$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$
180.19.5a	R	$5$	$1$	$5$	$5$	$-1$	$-1$	$-1$	$0$	$0$	$1$	$1$	$0$	$0$	$0$	$0$
180.19.5b1	C	$5$	$1$	$5{\zeta }_3^{-1}$	$5{\zeta }_3$	$-1$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$0$	$0$	${\zeta }_3$	${\zeta }_3^{-1}$	$0$	$0$	$0$	$0$
180.19.5b2	C	$5$	$1$	$5{\zeta }_3$	$5{\zeta }_3^{-1}$	$-1$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$0$	$0$	${\zeta }_3^{-1}$	${\zeta }_3$	$0$	$0$	$0$	$0$