LMFDB - Galois group: 30T47

Show commands: Magma

magma: G := TransitiveGroup(30, 47);

Group action invariants

Degree $n$:	$30$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$47$	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:	(1,26,11,30,3,25,10,29,2,27,12,28)(4,18,7,22,6,17,9,24,5,16,8,23)(13,19,14,20,15,21), (1,8,15,4,12,2,9,13,5,10,3,7,14,6,11)(16,23,29,21,25,18,22,28,20,27,17,24,30,19,26)	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 2: $C_2$
Degree 3: None
Degree 5: $F_5$
Degree 6: $S_3\times C_3$
Degree 10: $F_5$
Degree 15: None

Low degree siblings

45T18
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 $	$2$	$3$	$(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)$
	$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$2$	$3$	$(16,18,17)(19,21,20)(22,24,23)(25,27,26)(28,30,29)$
	$ 6, 6, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1 $	$10$	$6$	$( 4,14)( 5,15)( 6,13)( 7,11)( 8,12)( 9,10)(16,28,18,30,17,29)(19,26,21,25,20, 27)(22,23,24)$
	$ 6, 6, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1 $	$10$	$6$	$( 4,14)( 5,15)( 6,13)( 7,11)( 8,12)( 9,10)(16,29,17,30,18,28)(19,27,20,25,21, 26)(22,24,23)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $	$5$	$2$	$( 4,14)( 5,15)( 6,13)( 7,11)( 8,12)( 9,10)(16,30)(17,28)(18,29)(19,25)(20,26) (21,27)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$1$	$3$	$( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$2$	$3$	$( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,30,29)$
	$ 6, 6, 6, 6, 3, 3 $	$5$	$6$	$( 1, 2, 3)( 4,15, 6,14, 5,13)( 7,12, 9,11, 8,10)(16,28,18,30,17,29) (19,26,21,25,20,27)(22,23,24)$
	$ 6, 6, 6, 6, 3, 3 $	$10$	$6$	$( 1, 2, 3)( 4,15, 6,14, 5,13)( 7,12, 9,11, 8,10)(16,29,17,30,18,28) (19,27,20,25,21,26)(22,24,23)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$1$	$3$	$( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,30,29)$
	$ 6, 6, 6, 6, 3, 3 $	$5$	$6$	$( 1, 3, 2)( 4,13, 5,14, 6,15)( 7,10, 8,11, 9,12)(16,29,17,30,18,28) (19,27,20,25,21,26)(22,24,23)$
	$ 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 $	$4$	$15$	$( 1, 6, 7,10,13, 2, 4, 8,11,14, 3, 5, 9,12,15)(16,21,22,27,30)(17,19,23,25,28) (18,20,24,26,29)$
	$ 12, 12, 6 $	$15$	$12$	$( 1,16,13,26, 2,17,14,27, 3,18,15,25)( 4,22,12,20, 5,23,10,21, 6,24,11,19) ( 7,28, 9,30, 8,29)$
	$ 4, 4, 4, 4, 4, 4, 2, 2, 2 $	$15$	$4$	$( 1,16,14,27)( 2,17,15,25)( 3,18,13,26)( 4,22,10,21)( 5,23,11,19)( 6,24,12,20) ( 7,28)( 8,29)( 9,30)$
	$ 12, 12, 6 $	$15$	$12$	$( 1,16,15,25, 3,18,14,27, 2,17,13,26)( 4,22,11,19, 6,24,10,21, 5,23,12,20) ( 7,28, 8,29, 9,30)$
	$ 12, 12, 6 $	$15$	$12$	$( 1,16, 7,19, 3,18, 9,21, 2,17, 8,20)( 4,27, 5,25, 6,26)(10,30,15,23,12,29,14, 22,11,28,13,24)$
	$ 12, 12, 6 $	$15$	$12$	$( 1,16, 8,20, 2,17, 9,21, 3,18, 7,19)( 4,27, 6,26, 5,25)(10,30,13,24,11,28,14, 22,12,29,15,23)$
	$ 4, 4, 4, 4, 4, 4, 2, 2, 2 $	$15$	$4$	$( 1,16, 9,21)( 2,17, 7,19)( 3,18, 8,20)( 4,27)( 5,25)( 6,26)(10,30,14,22) (11,28,15,23)(12,29,13,24)$

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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Hilbert	Bianchi

Elliptic curves over $\Q$
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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	30T47
Degree	$30$
Order	$180$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_{15}:C_{12}$