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

$\card{(G/N)}$ 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 $\leq 47$

$\card{(G/N)}$	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	Index	Representative
1A	$1^{30}$	$1$	$1$	$0$	$()$
2A	$2^{12},1^{6}$	$5$	$2$	$12$	$( 1,10)( 2,11)( 3,12)( 4, 9)( 5, 7)( 6, 8)(16,22)(17,23)(18,24)(25,28)(26,29)(27,30)$
3A1	$3^{10}$	$1$	$3$	$20$	$( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)$
3A-1	$3^{10}$	$1$	$3$	$20$	$( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20)(22,24,23)(25,27,26)(28,30,29)$
3B	$3^{5},1^{15}$	$2$	$3$	$10$	$( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)$
3C1	$3^{10}$	$2$	$3$	$20$	$( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,18,17)(19,21,20)(22,24,23)(25,27,26)(28,30,29)$
3C-1	$3^{5},1^{15}$	$2$	$3$	$10$	$(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)$
4A1	$4^{6},2^{3}$	$15$	$4$	$21$	$( 1,28,10,25)( 2,29,11,26)( 3,30,12,27)( 4,23, 9,17)( 5,24, 7,18)( 6,22, 8,16)(13,21)(14,19)(15,20)$
4A-1	$4^{6},2^{3}$	$15$	$4$	$21$	$( 1,25,10,28)( 2,26,11,29)( 3,27,12,30)( 4,17, 9,23)( 5,18, 7,24)( 6,16, 8,22)(13,21)(14,19)(15,20)$
5A	$5^{6}$	$4$	$5$	$24$	$( 1,14,10, 9, 4)( 2,15,11, 7, 5)( 3,13,12, 8, 6)(16,30,27,22,21)(17,28,25,23,19)(18,29,26,24,20)$
6A1	$6^{4},3^{2}$	$5$	$6$	$24$	$( 1,11, 3,10, 2,12)( 4, 7, 6, 9, 5, 8)(13,14,15)(16,23,18,22,17,24)(19,20,21)(25,29,27,28,26,30)$
6A-1	$6^{4},3^{2}$	$5$	$6$	$24$	$( 1,12, 2,10, 3,11)( 4, 8, 5, 9, 6, 7)(13,15,14)(16,24,17,22,18,23)(19,21,20)(25,30,26,28,27,29)$
6B	$6^{4},3^{2}$	$10$	$6$	$24$	$( 1, 2, 3)( 4,15, 6,14, 5,13)( 7,12, 9,11, 8,10)(16,29,17,30,18,28)(19,27,20,25,21,26)(22,24,23)$
6C1	$6^{2},3,2^{6},1^{3}$	$10$	$6$	$18$	$( 1, 3, 2)( 4,13, 5,14, 6,15)( 7,10, 8,11, 9,12)(16,30)(17,28)(18,29)(19,25)(20,26)(21,27)$
6C-1	$6^{2},3,2^{6},1^{3}$	$10$	$6$	$18$	$( 4,14)( 5,15)( 6,13)( 7,11)( 8,12)( 9,10)(16,28,18,30,17,29)(19,26,21,25,20,27)(22,23,24)$
12A1	$12^{2},6$	$15$	$12$	$27$	$( 1,27,11,28, 3,26,10,30, 2,25,12,29)( 4,16, 7,23, 6,18, 9,22, 5,17, 8,24)(13,20,14,21,15,19)$
12A-1	$12^{2},6$	$15$	$12$	$27$	$( 1,29,12,25, 2,30,10,26, 3,28,11,27)( 4,24, 8,17, 5,22, 9,18, 6,23, 7,16)(13,19,15,21,14,20)$
12A5	$12^{2},6$	$15$	$12$	$27$	$( 1,26,12,28, 2,27,10,29, 3,25,11,30)( 4,18, 8,23, 5,16, 9,24, 6,17, 7,22)(13,19,15,21,14,20)$
12A-5	$12^{2},6$	$15$	$12$	$27$	$( 1,30,11,25, 3,29,10,27, 2,28,12,26)( 4,22, 7,17, 6,24, 9,16, 5,23, 8,18)(13,20,14,21,15,19)$
15A1	$15,5^{3}$	$4$	$15$	$26$	$( 1,12, 5,14, 8, 2,10, 6,15, 9, 3,11, 4,13, 7)(16,27,21,30,22)(17,25,19,28,23)(18,26,20,29,24)$
15A-1	$15^{2}$	$4$	$15$	$28$	$( 1,11, 6,14, 7, 3,10, 5,13, 9, 2,12, 4,15, 8)(16,26,19,30,24,17,27,20,28,22,18,25,21,29,23)$
15B1	$15,5^{3}$	$4$	$15$	$26$	$( 1, 4, 9,10,14)( 2, 5, 7,11,15)( 3, 6, 8,12,13)(16,19,24,27,28,18,21,23,26,30,17,20,22,25,29)$
15B-1	$15^{2}$	$4$	$15$	$28$	$( 1,15,12, 9, 5, 3,14,11, 8, 4, 2,13,10, 7, 6)(16,28,26,22,19,18,30,25,24,21,17,29,27,23,20)$
15C1	$15,5^{3}$	$4$	$15$	$26$	$( 1,10, 4,14, 9)( 2,11, 5,15, 7)( 3,12, 6,13, 8)(16,25,20,30,23,18,27,19,29,22,17,26,21,28,24)$
15C-1	$15^{2}$	$4$	$15$	$28$	$( 1, 5, 8,10,15, 3, 4, 7,12,14, 2, 6, 9,11,13)(16,20,23,27,29,17,21,24,25,30,18,19,22,26,28)$
15C2	$15,5^{3}$	$4$	$15$	$26$	$( 1, 6, 7,10,13, 2, 4, 8,11,14, 3, 5, 9,12,15)(16,21,22,27,30)(17,19,23,25,28)(18,20,24,26,29)$
15C-2	$15^{2}$	$4$	$15$	$28$	$( 1,13,11, 9, 6, 2,14,12, 7, 4, 3,15,10, 8, 5)(16,29,25,22,20,17,30,26,23,21,18,28,27,24,19)$

Malle's constant $a(G)$: $1/10$

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	15C1	15B1	15C-1	15A1	15C2	15B-1	15C-2	15A-1
	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	3C1	3A1
	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:	$C_{15}:C_{12}$