LMFDB - Galois group: 45T18

Show commands: Magma

magma: G := TransitiveGroup(45, 18);

Group action invariants

Degree $n$:	$45$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$18$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$C_{15}:C_{12}$
Parity:	$1$	magma: IsEven(G);
Primitive:	no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:	$3$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:	(1,9,15,19,27,31,38,45,4,10,18,23,30,35,40)(2,7,14,21,26,33,37,44,6,11,17,22,29,34,41)(3,8,13,20,25,32,39,43,5,12,16,24,28,36,42), (1,43,38,25)(2,45,37,27)(3,44,39,26)(7,12,17,28)(8,10,16,30)(9,11,18,29)(13,24,42,32)(14,22,41,33)(15,23,40,31)(19,36)(20,34)(21,35)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$
$3$: $C_3$
$4$: $C_4$
$6$: $S_3$, $C_6$
$12$: $C_{12}$, $C_3 : C_4$
$18$: $S_3\times C_3$
$20$: $F_5$
$36$: $C_3\times (C_3 : C_4)$
$60$: $C_{15} : C_4$, $F_5\times C_3$

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$4$:	$C_4$
$6$:	$S_3$, $C_6$
$12$:	$C_{12}$, $C_3 : C_4$
$18$:	$S_3\times C_3$
$20$:	$F_5$
$36$:	$C_3\times (C_3 : C_4)$
$60$:	$C_{15} : C_4$, $F_5\times C_3$

Subfields

Degree 3: $C_3$, $S_3$
Degree 5: $F_5$
Degree 9: $S_3\times C_3$
Degree 15: $C_{15} : C_4$, $F_5\times C_3$

Low degree siblings

30T47
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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1, 1, 1 $	$15$	$4$	$( 4, 7,15,26)( 5, 9,13,27)( 6, 8,14,25)(10,19,38,30)(11,21,37,29)(12,20,39,28) (16,33)(17,31)(18,32)(22,43,41,36)(23,44,40,34)(24,45,42,35)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$5$	$2$	$( 4,15)( 5,13)( 6,14)( 7,26)( 8,25)( 9,27)(10,38)(11,37)(12,39)(19,30)(20,28) (21,29)(22,41)(23,40)(24,42)(34,44)(35,45)(36,43)$
	$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1, 1, 1 $	$15$	$4$	$( 4,26,15, 7)( 5,27,13, 9)( 6,25,14, 8)(10,30,38,19)(11,29,37,21)(12,28,39,20) (16,33)(17,31)(18,32)(22,36,41,43)(23,34,40,44)(24,35,42,45)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$1$	$3$	$( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,11,12)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,30,29)(31,33,32)(34,36,35)(37,39,38)(40,41,42) (43,45,44)$
	$ 12, 12, 12, 6, 3 $	$15$	$12$	$( 1, 2, 3)( 4, 8,13,26, 6, 9,15,25, 5, 7,14,27)(10,21,39,30,11,20,38,29,12,19, 37,28)(16,32,17,33,18,31)(22,45,40,36,24,44,41,35,23,43,42,34)$
	$ 6, 6, 6, 6, 6, 6, 3, 3, 3 $	$5$	$6$	$( 1, 2, 3)( 4,14, 5,15, 6,13)( 7,25, 9,26, 8,27)(10,37,12,38,11,39)(16,18,17) (19,29,20,30,21,28)(22,42,23,41,24,40)(31,33,32)(34,43,35,44,36,45)$
	$ 12, 12, 12, 6, 3 $	$15$	$12$	$( 1, 2, 3)( 4,25,13, 7, 6,27,15, 8, 5,26,14, 9)(10,29,39,19,11,28,38,21,12,30, 37,20)(16,32,17,33,18,31)(22,35,40,43,24,34,41,45,23,36,42,44)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$1$	$3$	$( 1, 3, 2)( 4, 5, 6)( 7, 9, 8)(10,12,11)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36)(37,38,39)(40,42,41) (43,44,45)$
	$ 12, 12, 12, 6, 3 $	$15$	$12$	$( 1, 3, 2)( 4, 9,14,26, 5, 8,15,27, 6, 7,13,25)(10,20,37,30,12,21,38,28,11,19, 39,29)(16,31,18,33,17,32)(22,44,42,36,23,45,41,34,24,43,40,35)$
	$ 6, 6, 6, 6, 6, 6, 3, 3, 3 $	$5$	$6$	$( 1, 3, 2)( 4,13, 6,15, 5,14)( 7,27, 8,26, 9,25)(10,39,11,38,12,37)(16,17,18) (19,28,21,30,20,29)(22,40,24,41,23,42)(31,32,33)(34,45,36,44,35,43)$
	$ 12, 12, 12, 6, 3 $	$15$	$12$	$( 1, 3, 2)( 4,27,14, 7, 5,25,15, 9, 6,26,13, 8)(10,28,37,19,12,29,38,20,11,30, 39,21)(16,31,18,33,17,32)(22,34,42,43,23,35,41,44,24,36,40,45)$
	$ 15, 15, 15 $	$4$	$15$	$( 1, 4, 9,10,15,18,19,23,27,30,31,35,38,40,45)( 2, 6, 7,11,14,17,21,22,26,29, 33,34,37,41,44)( 3, 5, 8,12,13,16,20,24,25,28,32,36,39,42,43)$
	$ 6, 6, 6, 6, 6, 6, 3, 3, 3 $	$10$	$6$	$( 1, 4,18,19,31,35)( 2, 6,17,21,33,34)( 3, 5,16,20,32,36)( 7,29,22,44,37,14) ( 8,28,24,43,39,13)( 9,30,23,45,38,15)(10,40,27)(11,41,26)(12,42,25)$
	$ 15, 15, 15 $	$4$	$15$	$( 1, 5, 7,10,13,17,19,24,26,30,32,34,38,42,44)( 2, 4, 8,11,15,16,21,23,25,29, 31,36,37,40,43)( 3, 6, 9,12,14,18,20,22,27,28,33,35,39,41,45)$
	$ 6, 6, 6, 6, 6, 6, 3, 3, 3 $	$10$	$6$	$( 1, 5,17,19,32,34)( 2, 4,16,21,31,36)( 3, 6,18,20,33,35)( 7,30,24,44,38,13) ( 8,29,23,43,37,15)( 9,28,22,45,39,14)(10,42,26)(11,40,25)(12,41,27)$
	$ 15, 15, 15 $	$4$	$15$	$( 1, 6, 8,10,14,16,19,22,25,30,33,36,38,41,43)( 2, 5, 9,11,13,18,21,24,27,29, 32,35,37,42,45)( 3, 4, 7,12,15,17,20,23,26,28,31,34,39,40,44)$
	$ 6, 6, 6, 6, 6, 6, 3, 3, 3 $	$10$	$6$	$( 1, 6,16,19,33,36)( 2, 5,18,21,32,35)( 3, 4,17,20,31,34)( 7,28,23,44,39,15) ( 8,30,22,43,38,14)( 9,29,24,45,37,13)(10,41,25)(11,42,27)(12,40,26)$
	$ 5, 5, 5, 5, 5, 5, 5, 5, 5 $	$4$	$5$	$( 1,10,19,30,38)( 2,11,21,29,37)( 3,12,20,28,39)( 4,15,23,31,40) ( 5,13,24,32,42)( 6,14,22,33,41)( 7,17,26,34,44)( 8,16,25,36,43) ( 9,18,27,35,45)$
	$ 15, 15, 15 $	$4$	$15$	$( 1,11,20,30,37, 3,10,21,28,38, 2,12,19,29,39)( 4,14,24,31,41, 5,15,22,32,40, 6,13,23,33,42)( 7,16,27,34,43, 9,17,25,35,44, 8,18,26,36,45)$
	$ 15, 15, 15 $	$4$	$15$	$( 1,12,21,30,39, 2,10,20,29,38, 3,11,19,28,37)( 4,13,22,31,42, 6,15,24,33,40, 5,14,23,32,41)( 7,18,25,34,45, 8,17,27,36,44, 9,16,26,35,43)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$2$	$3$	$( 1,16,33)( 2,18,32)( 3,17,31)( 4,20,34)( 5,21,35)( 6,19,36)( 7,23,39) ( 8,22,38)( 9,24,37)(10,25,41)(11,27,42)(12,26,40)(13,29,45)(14,30,43) (15,28,44)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$2$	$3$	$( 1,17,32)( 2,16,31)( 3,18,33)( 4,21,36)( 5,19,34)( 6,20,35)( 7,24,38) ( 8,23,37)( 9,22,39)(10,26,42)(11,25,40)(12,27,41)(13,30,44)(14,28,45) (15,29,43)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$2$	$3$	$( 1,18,31)( 2,17,33)( 3,16,32)( 4,19,35)( 5,20,36)( 6,21,34)( 7,22,37) ( 8,24,39)( 9,23,38)(10,27,40)(11,26,41)(12,25,42)(13,28,43)(14,29,44) (15,30,45)$
	$ 15, 15, 15 $	$4$	$15$	$( 1,22,43,19,41,16,38,14,36,10,33, 8,30, 6,25)( 2,24,45,21,42,18,37,13,35,11, 32, 9,29, 5,27)( 3,23,44,20,40,17,39,15,34,12,31, 7,28, 4,26)$
	$ 15, 15, 15 $	$4$	$15$	$( 1,23,45,19,40,18,38,15,35,10,31, 9,30, 4,27)( 2,22,44,21,41,17,37,14,34,11, 33, 7,29, 6,26)( 3,24,43,20,42,16,39,13,36,12,32, 8,28, 5,25)$
	$ 15, 15, 15 $	$4$	$15$	$( 1,24,44,19,42,17,38,13,34,10,32, 7,30, 5,26)( 2,23,43,21,40,16,37,15,36,11, 31, 8,29, 4,25)( 3,22,45,20,41,18,39,14,35,12,33, 9,28, 6,27)$

magma: ConjugacyClasses(G);

Group invariants

Order:	$180=2^{2} \cdot 3^{2} \cdot 5$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	not nilpotent
Label:	180.21	magma: IdentifyGroup(G);
Character table:

		1A	2A	3A1	3A-1	3B	3C1	3C-1	4A1	4A-1	5A	6A1	6A-1	6B	6C1	6C-1	12A1	12A-1	12A5	12A-5	15A1	15A-1	15B1	15B-1	15C1	15C-1	15C2	15C-2
	Size	1	5	1	1	2	2	2	15	15	4	5	5	10	10	10	15	15	15	15	4	4	4	4	4	4	4	4
	2 P	1A	1A	3A-1	3A1	3C1	3B	3C-1	2A	2A	5A	3A-1	3A1	3B	3C1	3C-1	6A1	6A-1	6A-1	6A1	15C-1	15B1	15C-2	15A1	15C1	15B-1	15A-1	15C2
	3 P	1A	2A	1A	1A	1A	1A	1A	4A-1	4A1	5A	2A	2A	2A	2A	2A	4A1	4A-1	4A1	4A-1	5A	5A	5A	5A	5A	5A	5A	5A
	5 P	1A	2A	3A-1	3A1	3C1	3B	3C-1	4A1	4A-1	1A	6A-1	6A1	6B	6C-1	6C1	12A5	12A-5	12A1	12A-1	3C1	3B	3C-1	3A-1	3C-1	3B	3A1	3C1
	Type
180.21.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
180.21.1b	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
180.21.1c1	C	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$
180.21.1c2	C	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$
180.21.1d1	C	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$- i$	$i$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$i$	$- i$	$i$	$- i$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
180.21.1d2	C	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$i$	$- i$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- i$	$i$	$- i$	$i$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
180.21.1e1	C	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$- 1$	$- 1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$
180.21.1e2	C	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$- 1$	$- 1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$
180.21.1f1	C	$1$	$- 1$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$	$1$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$	$- ζ_{12}^{3}$	$ζ_{12}^{3}$	$1$	$ζ_{12}^{2}$	$- ζ_{12}^{4}$	$- 1$	$ζ_{12}^{2}$	$- ζ_{12}^{4}$	$- ζ_{12}$	$ζ_{12}^{5}$	$- ζ_{12}^{5}$	$ζ_{12}$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$	$1$	$1$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$
180.21.1f2	C	$1$	$- 1$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$	$1$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$	$ζ_{12}^{3}$	$- ζ_{12}^{3}$	$1$	$- ζ_{12}^{4}$	$ζ_{12}^{2}$	$- 1$	$- ζ_{12}^{4}$	$ζ_{12}^{2}$	$ζ_{12}^{5}$	$- ζ_{12}$	$ζ_{12}$	$- ζ_{12}^{5}$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$	$1$	$1$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$
180.21.1f3	C	$1$	$- 1$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$	$1$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$	$ζ_{12}^{3}$	$- ζ_{12}^{3}$	$1$	$ζ_{12}^{2}$	$- ζ_{12}^{4}$	$- 1$	$ζ_{12}^{2}$	$- ζ_{12}^{4}$	$ζ_{12}$	$- ζ_{12}^{5}$	$ζ_{12}^{5}$	$- ζ_{12}$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$	$1$	$1$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$
180.21.1f4	C	$1$	$- 1$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$	$1$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$	$- ζ_{12}^{3}$	$ζ_{12}^{3}$	$1$	$- ζ_{12}^{4}$	$ζ_{12}^{2}$	$- 1$	$- ζ_{12}^{4}$	$ζ_{12}^{2}$	$- ζ_{12}^{5}$	$ζ_{12}$	$- ζ_{12}$	$ζ_{12}^{5}$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$	$1$	$1$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$
180.21.2a	R	$2$	$2$	$2$	$2$	$- 1$	$- 1$	$- 1$	$0$	$0$	$2$	$2$	$2$	$- 1$	$- 1$	$- 1$	$0$	$0$	$0$	$0$	$2$	$2$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$
180.21.2b	S	$2$	$- 2$	$2$	$2$	$- 1$	$- 1$	$- 1$	$0$	$0$	$2$	$- 2$	$- 2$	$1$	$1$	$1$	$0$	$0$	$0$	$0$	$2$	$2$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$
180.21.2c1	C	$2$	$2$	$2 ζ_{3}^{- 1}$	$2 ζ_{3}$	$- 1$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$0$	$0$	$2$	$2 ζ_{3}^{- 1}$	$2 ζ_{3}$	$- 1$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$0$	$0$	$0$	$0$	$2 ζ_{3}$	$2 ζ_{3}^{- 1}$	$- 1$	$- 1$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$
180.21.2c2	C	$2$	$2$	$2 ζ_{3}$	$2 ζ_{3}^{- 1}$	$- 1$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$0$	$0$	$2$	$2 ζ_{3}$	$2 ζ_{3}^{- 1}$	$- 1$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$0$	$0$	$0$	$0$	$2 ζ_{3}^{- 1}$	$2 ζ_{3}$	$- 1$	$- 1$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$
180.21.2d1	C	$2$	$- 2$	$2 ζ_{3}^{- 1}$	$2 ζ_{3}$	$- 1$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$0$	$0$	$2$	$- 2 ζ_{3}^{- 1}$	$- 2 ζ_{3}$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$0$	$0$	$0$	$0$	$2 ζ_{3}$	$2 ζ_{3}^{- 1}$	$- 1$	$- 1$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$
180.21.2d2	C	$2$	$- 2$	$2 ζ_{3}$	$2 ζ_{3}^{- 1}$	$- 1$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$0$	$0$	$2$	$- 2 ζ_{3}$	$- 2 ζ_{3}^{- 1}$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$0$	$0$	$0$	$0$	$2 ζ_{3}^{- 1}$	$2 ζ_{3}$	$- 1$	$- 1$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$
180.21.4a	R	$4$	$0$	$4$	$4$	$4$	$4$	$4$	$0$	$0$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$
180.21.4b1	C	$4$	$0$	$4$	$4$	$- 2$	$- 2$	$- 2$	$0$	$0$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$	$- 1$	$2 - 2 ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - 2 ζ_{15}^{4} + ζ_{15}^{5} - ζ_{15}^{7}$	$- 1 + 2 ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + 2 ζ_{15}^{4} - ζ_{15}^{5} + ζ_{15}^{7}$	$2 - 2 ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - 2 ζ_{15}^{4} + ζ_{15}^{5} - ζ_{15}^{7}$	$- 1 + 2 ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + 2 ζ_{15}^{4} - ζ_{15}^{5} + ζ_{15}^{7}$	$2 - 2 ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - 2 ζ_{15}^{4} + ζ_{15}^{5} - ζ_{15}^{7}$	$- 1 + 2 ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + 2 ζ_{15}^{4} - ζ_{15}^{5} + ζ_{15}^{7}$
180.21.4b2	C	$4$	$0$	$4$	$4$	$- 2$	$- 2$	$- 2$	$0$	$0$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$	$- 1$	$- 1 + 2 ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + 2 ζ_{15}^{4} - ζ_{15}^{5} + ζ_{15}^{7}$	$2 - 2 ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - 2 ζ_{15}^{4} + ζ_{15}^{5} - ζ_{15}^{7}$	$- 1 + 2 ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + 2 ζ_{15}^{4} - ζ_{15}^{5} + ζ_{15}^{7}$	$2 - 2 ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - 2 ζ_{15}^{4} + ζ_{15}^{5} - ζ_{15}^{7}$	$- 1 + 2 ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + 2 ζ_{15}^{4} - ζ_{15}^{5} + ζ_{15}^{7}$	$2 - 2 ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - 2 ζ_{15}^{4} + ζ_{15}^{5} - ζ_{15}^{7}$
180.21.4c1	C	$4$	$0$	$4 ζ_{3}^{- 1}$	$4 ζ_{3}$	$4$	$4 ζ_{3}$	$4 ζ_{3}^{- 1}$	$0$	$0$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$- 1$	$- 1$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$
180.21.4c2	C	$4$	$0$	$4 ζ_{3}$	$4 ζ_{3}^{- 1}$	$4$	$4 ζ_{3}^{- 1}$	$4 ζ_{3}$	$0$	$0$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$- 1$	$- 1$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$
180.21.4d1	C	$4$	$0$	$4 ζ_{15}^{- 5}$	$4 ζ_{15}^{5}$	$- 2$	$- 2 ζ_{15}^{5}$	$- 2 ζ_{15}^{- 5}$	$0$	$0$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- ζ_{15}^{5}$	$- ζ_{15}^{- 5}$	$2 - 2 ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - 2 ζ_{15}^{4} + ζ_{15}^{5} - ζ_{15}^{7}$	$- 1 + 2 ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + 2 ζ_{15}^{4} - ζ_{15}^{5} + ζ_{15}^{7}$	$- 2 + ζ_{15} + 2 ζ_{15}^{2} - 2 ζ_{15}^{3} + ζ_{15}^{4} - ζ_{15}^{5} + 2 ζ_{15}^{7}$	$- ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} - ζ_{15}^{4} + ζ_{15}^{5} + ζ_{15}^{7}$	$ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} + ζ_{15}^{4} - ζ_{15}^{7}$	$1 - ζ_{15} - 2 ζ_{15}^{2} + 2 ζ_{15}^{3} - ζ_{15}^{4} - 2 ζ_{15}^{7}$
180.21.4d2	C	$4$	$0$	$4 ζ_{15}^{5}$	$4 ζ_{15}^{- 5}$	$- 2$	$- 2 ζ_{15}^{- 5}$	$- 2 ζ_{15}^{5}$	$0$	$0$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- ζ_{15}^{- 5}$	$- ζ_{15}^{5}$	$- 1 + 2 ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + 2 ζ_{15}^{4} - ζ_{15}^{5} + ζ_{15}^{7}$	$2 - 2 ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - 2 ζ_{15}^{4} + ζ_{15}^{5} - ζ_{15}^{7}$	$- ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} - ζ_{15}^{4} + ζ_{15}^{5} + ζ_{15}^{7}$	$- 2 + ζ_{15} + 2 ζ_{15}^{2} - 2 ζ_{15}^{3} + ζ_{15}^{4} - ζ_{15}^{5} + 2 ζ_{15}^{7}$	$1 - ζ_{15} - 2 ζ_{15}^{2} + 2 ζ_{15}^{3} - ζ_{15}^{4} - 2 ζ_{15}^{7}$	$ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} + ζ_{15}^{4} - ζ_{15}^{7}$
180.21.4d3	C	$4$	$0$	$4 ζ_{15}^{- 5}$	$4 ζ_{15}^{5}$	$- 2$	$- 2 ζ_{15}^{5}$	$- 2 ζ_{15}^{- 5}$	$0$	$0$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- ζ_{15}^{5}$	$- ζ_{15}^{- 5}$	$- 1 + 2 ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + 2 ζ_{15}^{4} - ζ_{15}^{5} + ζ_{15}^{7}$	$2 - 2 ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - 2 ζ_{15}^{4} + ζ_{15}^{5} - ζ_{15}^{7}$	$1 - ζ_{15} - 2 ζ_{15}^{2} + 2 ζ_{15}^{3} - ζ_{15}^{4} - 2 ζ_{15}^{7}$	$ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} + ζ_{15}^{4} - ζ_{15}^{7}$	$- ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} - ζ_{15}^{4} + ζ_{15}^{5} + ζ_{15}^{7}$	$- 2 + ζ_{15} + 2 ζ_{15}^{2} - 2 ζ_{15}^{3} + ζ_{15}^{4} - ζ_{15}^{5} + 2 ζ_{15}^{7}$
180.21.4d4	C	$4$	$0$	$4 ζ_{15}^{5}$	$4 ζ_{15}^{- 5}$	$- 2$	$- 2 ζ_{15}^{- 5}$	$- 2 ζ_{15}^{5}$	$0$	$0$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- ζ_{15}^{- 5}$	$- ζ_{15}^{5}$	$2 - 2 ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} - 2 ζ_{15}^{4} + ζ_{15}^{5} - ζ_{15}^{7}$	$- 1 + 2 ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} + 2 ζ_{15}^{4} - ζ_{15}^{5} + ζ_{15}^{7}$	$ζ_{15} - ζ_{15}^{2} + ζ_{15}^{3} + ζ_{15}^{4} - ζ_{15}^{7}$	$1 - ζ_{15} - 2 ζ_{15}^{2} + 2 ζ_{15}^{3} - ζ_{15}^{4} - 2 ζ_{15}^{7}$	$- 2 + ζ_{15} + 2 ζ_{15}^{2} - 2 ζ_{15}^{3} + ζ_{15}^{4} - ζ_{15}^{5} + 2 ζ_{15}^{7}$	$- ζ_{15} + ζ_{15}^{2} - ζ_{15}^{3} - ζ_{15}^{4} + ζ_{15}^{5} + ζ_{15}^{7}$

magma: CharacterTable(G);

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	45T18
Degree	$45$
Order	$180$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_{15}:C_{12}$