LMFDB - Galois group: 30T26

Show commands: Magma

magma: G := TransitiveGroup(30, 26);

Group action invariants

Degree $n$:	$30$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$26$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$C_6\times F_5$
Parity:	$-1$	magma: IsEven(G);
Primitive:	no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:	$6$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:	(1,2)(3,27,9,15)(4,28,10,16)(5,23,17,30)(6,24,18,29)(7,20,25,13)(8,19,26,14)(11,12)(21,22), (1,10,5,7,21,29,25,27,11,19,15,17)(2,9,6,8,22,30,26,28,12,20,16,18)(3,23,13)(4,24,14)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$ x 3
$3$: $C_3$
$4$: $C_4$ x 2, $C_2^2$
$6$: $C_6$ x 3
$8$: $C_4\times C_2$
$12$: $C_{12}$ x 2, $C_6\times C_2$
$20$: $F_5$
$24$: 24T2
$40$: $F_{5}\times C_2$
$60$: $F_5\times C_3$

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$3$:	$C_3$
$4$:	$C_4$ x 2, $C_2^2$
$6$:	$C_6$ x 3
$8$:	$C_4\times C_2$
$12$:	$C_{12}$ x 2, $C_6\times C_2$
$20$:	$F_5$
$24$:	24T2
$40$:	$F_{5}\times C_2$
$60$:	$F_5\times C_3$

Subfields

Degree 2: $C_2$
Degree 3: $C_3$
Degree 5: $F_5$
Degree 6: $C_6$
Degree 10: $F_{5}\times C_2$
Degree 15: $F_5\times C_3$

Low degree siblings

30T26
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 $	$1$	$1$	$()$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $	$5$	$2$	$( 3, 9)( 4,10)( 5,17)( 6,18)( 7,25)( 8,26)(13,20)(14,19)(15,27)(16,28)(23,30) (24,29)$
	$ 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1 $	$5$	$4$	$( 3,16, 9,28)( 4,15,10,27)( 5,29,17,24)( 6,30,18,23)( 7,14,25,19)( 8,13,26,20)$
	$ 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1 $	$5$	$4$	$( 3,28, 9,16)( 4,27,10,15)( 5,24,17,29)( 6,23,18,30)( 7,19,25,14)( 8,20,26,13)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$1$	$2$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$5$	$2$	$( 1, 2)( 3,10)( 4, 9)( 5,18)( 6,17)( 7,26)( 8,25)(11,12)(13,19)(14,20)(15,28) (16,27)(21,22)(23,29)(24,30)$
	$ 4, 4, 4, 4, 4, 4, 2, 2, 2 $	$5$	$4$	$( 1, 2)( 3,15, 9,27)( 4,16,10,28)( 5,30,17,23)( 6,29,18,24)( 7,13,25,20) ( 8,14,26,19)(11,12)(21,22)$
	$ 4, 4, 4, 4, 4, 4, 2, 2, 2 $	$5$	$4$	$( 1, 2)( 3,27, 9,15)( 4,28,10,16)( 5,23,17,30)( 6,24,18,29)( 7,20,25,13) ( 8,19,26,14)(11,12)(21,22)$
	$ 30 $	$4$	$30$	$( 1, 3, 5, 8,10,12,14,16,17,20,21,23,25,28,29, 2, 4, 6, 7, 9,11,13,15,18,19, 22,24,26,27,30)$
	$ 6, 6, 6, 6, 6 $	$5$	$6$	$( 1, 3,11,13,21,23)( 2, 4,12,14,22,24)( 5,20,15,30,25, 9)( 6,19,16,29,26,10) ( 7,28,17, 8,27,18)$
	$ 12, 12, 6 $	$5$	$12$	$( 1, 3,17,26,21,23, 7,16,11,13,27, 6)( 2, 4,18,25,22,24, 8,15,12,14,28, 5) ( 9,29,20,10,30,19)$
	$ 12, 12, 6 $	$5$	$12$	$( 1, 3,29, 8,21,23,19,28,11,13,10,18)( 2, 4,30, 7,22,24,20,27,12,14, 9,17) ( 5,26,15, 6,25,16)$
	$ 15, 15 $	$4$	$15$	$( 1, 4, 5, 7,10,11,14,15,17,19,21,24,25,27,29)( 2, 3, 6, 8, 9,12,13,16,18,20, 22,23,26,28,30)$
	$ 6, 6, 6, 6, 3, 3 $	$5$	$6$	$( 1, 4,11,14,21,24)( 2, 3,12,13,22,23)( 5,19,15,29,25,10)( 6,20,16,30,26, 9) ( 7,27,17)( 8,28,18)$
	$ 12, 12, 3, 3 $	$5$	$12$	$( 1, 4,17,25,21,24, 7,15,11,14,27, 5)( 2, 3,18,26,22,23, 8,16,12,13,28, 6) ( 9,30,20)(10,29,19)$
	$ 12, 12, 3, 3 $	$5$	$12$	$( 1, 4,29, 7,21,24,19,27,11,14,10,17)( 2, 3,30, 8,22,23,20,28,12,13, 9,18) ( 5,25,15)( 6,26,16)$
	$ 12, 12, 3, 3 $	$5$	$12$	$( 1, 5,27,14,11,15, 7,24,21,25,17, 4)( 2, 6,28,13,12,16, 8,23,22,26,18, 3) ( 9,20,30)(10,19,29)$
	$ 15, 15 $	$4$	$15$	$( 1, 5,10,14,17,21,25,29, 4, 7,11,15,19,24,27)( 2, 6, 9,13,18,22,26,30, 3, 8, 12,16,20,23,28)$
	$ 6, 6, 6, 6, 3, 3 $	$5$	$6$	$( 1, 5,21,25,11,15)( 2, 6,22,26,12,16)( 3,13,23)( 4,14,24)( 7,29,27,19,17,10) ( 8,30,28,20,18, 9)$
	$ 12, 12, 3, 3 $	$5$	$12$	$( 1, 5, 4,19,11,15,14,29,21,25,24,10)( 2, 6, 3,20,12,16,13,30,22,26,23, 9) ( 7,17,27)( 8,18,28)$
	$ 12, 12, 6 $	$5$	$12$	$( 1, 6,27,13,11,16, 7,23,21,26,17, 3)( 2, 5,28,14,12,15, 8,24,22,25,18, 4) ( 9,19,30,10,20,29)$
	$ 30 $	$4$	$30$	$( 1, 6,10,13,17,22,25,30, 4, 8,11,16,19,23,27, 2, 5, 9,14,18,21,26,29, 3, 7, 12,15,20,24,28)$
	$ 6, 6, 6, 6, 6 $	$5$	$6$	$( 1, 6,21,26,11,16)( 2, 5,22,25,12,15)( 3,14,23, 4,13,24)( 7,30,27,20,17, 9) ( 8,29,28,19,18,10)$
	$ 12, 12, 6 $	$5$	$12$	$( 1, 6, 4,20,11,16,14,30,21,26,24, 9)( 2, 5, 3,19,12,15,13,29,22,25,23,10) ( 7,18,27, 8,17,28)$
	$ 5, 5, 5, 5, 5, 5 $	$4$	$5$	$( 1, 7,14,19,25)( 2, 8,13,20,26)( 3, 9,16,22,28)( 4,10,15,21,27) ( 5,11,17,24,29)( 6,12,18,23,30)$
	$ 10, 10, 10 $	$4$	$10$	$( 1, 8,14,20,25, 2, 7,13,19,26)( 3,10,16,21,28, 4, 9,15,22,27)( 5,12,17,23,29, 6,11,18,24,30)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$1$	$3$	$( 1,11,21)( 2,12,22)( 3,13,23)( 4,14,24)( 5,15,25)( 6,16,26)( 7,17,27) ( 8,18,28)( 9,20,30)(10,19,29)$
	$ 6, 6, 6, 6, 6 $	$1$	$6$	$( 1,12,21, 2,11,22)( 3,14,23, 4,13,24)( 5,16,25, 6,15,26)( 7,18,27, 8,17,28) ( 9,19,30,10,20,29)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$1$	$3$	$( 1,21,11)( 2,22,12)( 3,23,13)( 4,24,14)( 5,25,15)( 6,26,16)( 7,27,17) ( 8,28,18)( 9,30,20)(10,29,19)$
	$ 6, 6, 6, 6, 6 $	$1$	$6$	$( 1,22,11, 2,21,12)( 3,24,13, 4,23,14)( 5,26,15, 6,25,16)( 7,28,17, 8,27,18) ( 9,29,20,10,30,19)$

magma: ConjugacyClasses(G);

Group invariants

Order:	$120=2^{3} \cdot 3 \cdot 5$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	not nilpotent
Label:	120.40	magma: IdentifyGroup(G);
Character table:

		1A	2A	2B	2C	3A1	3A-1	4A1	4A-1	4B1	4B-1	5A	6A1	6A-1	6B1	6B-1	6C1	6C-1	10A	12A1	12A-1	12A5	12A-5	12B1	12B-1	12B5	12B-5	15A1	15A-1	30A1	30A-1
	Size	1	1	5	5	1	1	5	5	5	5	4	1	1	5	5	5	5	4	5	5	5	5	5	5	5	5	4	4	4	4
	2 P	1A	1A	1A	1A	3A-1	3A1	2B	2B	2B	2B	5A	3A1	3A-1	3A-1	3A1	3A-1	3A1	5A	6B1	6B1	6B-1	6B1	6B-1	6B-1	6B1	6B-1	15A-1	15A1	15A1	15A-1
	3 P	1A	2A	2B	2C	1A	1A	4A-1	4A1	4B-1	4B1	5A	2A	2A	2C	2C	2B	2B	10A	4B-1	4B1	4A1	4A-1	4B1	4B-1	4A1	4A-1	5A	5A	10A	10A
	5 P	1A	2A	2B	2C	3A-1	3A1	4A1	4A-1	4B1	4B-1	1A	6A-1	6A1	6C1	6C-1	6B1	6B-1	2A	12B-1	12B5	12A1	12A-1	12B1	12B-5	12A5	12A-5	3A1	3A-1	6A1	6A-1
	Type
120.40.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
120.40.1b	R	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$
120.40.1c	R	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$
120.40.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$
120.40.1e1	C	$1$	$1$	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$1$	$1$	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$
120.40.1e2	C	$1$	$1$	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$1$	$1$	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$
120.40.1f1	C	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- i$	$i$	$- i$	$i$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$i$	$- i$	$i$	$- i$	$i$	$- i$	$i$	$- i$	$1$	$1$	$- 1$	$- 1$
120.40.1f2	C	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$i$	$- i$	$i$	$- i$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- i$	$i$	$- i$	$i$	$- i$	$i$	$- i$	$i$	$1$	$1$	$- 1$	$- 1$
120.40.1g1	C	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- i$	$i$	$i$	$- i$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$i$	$- i$	$i$	$- i$	$- i$	$i$	$- i$	$i$	$1$	$1$	$1$	$1$
120.40.1g2	C	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$i$	$- i$	$- i$	$i$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$- i$	$i$	$- i$	$i$	$i$	$- i$	$i$	$- i$	$1$	$1$	$1$	$1$
120.40.1h1	C	$1$	$- 1$	$1$	$- 1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$- 1$	$- 1$	$1$	$1$	$1$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$- 1$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$
120.40.1h2	C	$1$	$- 1$	$1$	$- 1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$- 1$	$- 1$	$1$	$1$	$1$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$- 1$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$
120.40.1i1	C	$1$	$- 1$	$1$	$- 1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$1$	$- 1$	$- 1$	$1$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$- 1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$
120.40.1i2	C	$1$	$- 1$	$1$	$- 1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$1$	$- 1$	$- 1$	$1$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$- 1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$
120.40.1j1	C	$1$	$1$	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$
120.40.1j2	C	$1$	$1$	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$
120.40.1k1	C	$1$	$- 1$	$- 1$	$1$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$	$- ζ_{12}^{3}$	$ζ_{12}^{3}$	$- ζ_{12}^{3}$	$ζ_{12}^{3}$	$1$	$- ζ_{12}^{4}$	$ζ_{12}^{2}$	$- ζ_{12}^{4}$	$ζ_{12}^{2}$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$	$- 1$	$- ζ_{12}^{5}$	$ζ_{12}$	$- ζ_{12}$	$ζ_{12}^{5}$	$- ζ_{12}^{5}$	$ζ_{12}$	$- ζ_{12}$	$ζ_{12}^{5}$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$	$ζ_{12}^{2}$	$- ζ_{12}^{4}$
120.40.1k2	C	$1$	$- 1$	$- 1$	$1$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$	$ζ_{12}^{3}$	$- ζ_{12}^{3}$	$ζ_{12}^{3}$	$- ζ_{12}^{3}$	$1$	$ζ_{12}^{2}$	$- ζ_{12}^{4}$	$ζ_{12}^{2}$	$- ζ_{12}^{4}$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$	$- 1$	$ζ_{12}$	$- ζ_{12}^{5}$	$ζ_{12}^{5}$	$- ζ_{12}$	$ζ_{12}$	$- ζ_{12}^{5}$	$ζ_{12}^{5}$	$- ζ_{12}$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$	$- ζ_{12}^{4}$	$ζ_{12}^{2}$
120.40.1k3	C	$1$	$- 1$	$- 1$	$1$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$	$ζ_{12}^{3}$	$- ζ_{12}^{3}$	$ζ_{12}^{3}$	$- ζ_{12}^{3}$	$1$	$- ζ_{12}^{4}$	$ζ_{12}^{2}$	$- ζ_{12}^{4}$	$ζ_{12}^{2}$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$	$- 1$	$ζ_{12}^{5}$	$- ζ_{12}$	$ζ_{12}$	$- ζ_{12}^{5}$	$ζ_{12}^{5}$	$- ζ_{12}$	$ζ_{12}$	$- ζ_{12}^{5}$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$	$ζ_{12}^{2}$	$- ζ_{12}^{4}$
120.40.1k4	C	$1$	$- 1$	$- 1$	$1$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$	$- ζ_{12}^{3}$	$ζ_{12}^{3}$	$- ζ_{12}^{3}$	$ζ_{12}^{3}$	$1$	$ζ_{12}^{2}$	$- ζ_{12}^{4}$	$ζ_{12}^{2}$	$- ζ_{12}^{4}$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$	$- 1$	$- ζ_{12}$	$ζ_{12}^{5}$	$- ζ_{12}^{5}$	$ζ_{12}$	$- ζ_{12}$	$ζ_{12}^{5}$	$- ζ_{12}^{5}$	$ζ_{12}$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$	$- ζ_{12}^{4}$	$ζ_{12}^{2}$
120.40.1l1	C	$1$	$1$	$- 1$	$- 1$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$	$- ζ_{12}^{3}$	$ζ_{12}^{3}$	$ζ_{12}^{3}$	$- ζ_{12}^{3}$	$1$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$	$- ζ_{12}^{4}$	$ζ_{12}^{2}$	$- ζ_{12}^{4}$	$ζ_{12}^{2}$	$1$	$- ζ_{12}^{5}$	$ζ_{12}$	$- ζ_{12}$	$ζ_{12}^{5}$	$ζ_{12}^{5}$	$- ζ_{12}$	$ζ_{12}$	$- ζ_{12}^{5}$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$
120.40.1l2	C	$1$	$1$	$- 1$	$- 1$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$	$ζ_{12}^{3}$	$- ζ_{12}^{3}$	$- ζ_{12}^{3}$	$ζ_{12}^{3}$	$1$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$	$ζ_{12}^{2}$	$- ζ_{12}^{4}$	$ζ_{12}^{2}$	$- ζ_{12}^{4}$	$1$	$ζ_{12}$	$- ζ_{12}^{5}$	$ζ_{12}^{5}$	$- ζ_{12}$	$- ζ_{12}$	$ζ_{12}^{5}$	$- ζ_{12}^{5}$	$ζ_{12}$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$
120.40.1l3	C	$1$	$1$	$- 1$	$- 1$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$	$ζ_{12}^{3}$	$- ζ_{12}^{3}$	$- ζ_{12}^{3}$	$ζ_{12}^{3}$	$1$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$	$- ζ_{12}^{4}$	$ζ_{12}^{2}$	$- ζ_{12}^{4}$	$ζ_{12}^{2}$	$1$	$ζ_{12}^{5}$	$- ζ_{12}$	$ζ_{12}$	$- ζ_{12}^{5}$	$- ζ_{12}^{5}$	$ζ_{12}$	$- ζ_{12}$	$ζ_{12}^{5}$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$
120.40.1l4	C	$1$	$1$	$- 1$	$- 1$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$	$- ζ_{12}^{3}$	$ζ_{12}^{3}$	$ζ_{12}^{3}$	$- ζ_{12}^{3}$	$1$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$	$ζ_{12}^{2}$	$- ζ_{12}^{4}$	$ζ_{12}^{2}$	$- ζ_{12}^{4}$	$1$	$- ζ_{12}$	$ζ_{12}^{5}$	$- ζ_{12}^{5}$	$ζ_{12}$	$ζ_{12}$	$- ζ_{12}^{5}$	$ζ_{12}^{5}$	$- ζ_{12}$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$
120.40.4a	R	$4$	$4$	$0$	$0$	$4$	$4$	$0$	$0$	$0$	$0$	$- 1$	$4$	$4$	$0$	$0$	$0$	$0$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$	$- 1$	$- 1$	$- 1$
120.40.4b	R	$4$	$- 4$	$0$	$0$	$4$	$4$	$0$	$0$	$0$	$0$	$- 1$	$- 4$	$- 4$	$0$	$0$	$0$	$0$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$	$- 1$	$1$	$1$
120.40.4c1	C	$4$	$4$	$0$	$0$	$4 ζ_{3}^{- 1}$	$4 ζ_{3}$	$0$	$0$	$0$	$0$	$- 1$	$4 ζ_{3}$	$4 ζ_{3}^{- 1}$	$0$	$0$	$0$	$0$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$
120.40.4c2	C	$4$	$4$	$0$	$0$	$4 ζ_{3}$	$4 ζ_{3}^{- 1}$	$0$	$0$	$0$	$0$	$- 1$	$4 ζ_{3}^{- 1}$	$4 ζ_{3}$	$0$	$0$	$0$	$0$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$
120.40.4d1	C	$4$	$- 4$	$0$	$0$	$4 ζ_{3}^{- 1}$	$4 ζ_{3}$	$0$	$0$	$0$	$0$	$- 1$	$- 4 ζ_{3}$	$- 4 ζ_{3}^{- 1}$	$0$	$0$	$0$	$0$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$
120.40.4d2	C	$4$	$- 4$	$0$	$0$	$4 ζ_{3}$	$4 ζ_{3}^{- 1}$	$0$	$0$	$0$	$0$	$- 1$	$- 4 ζ_{3}^{- 1}$	$- 4 ζ_{3}$	$0$	$0$	$0$	$0$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$

magma: CharacterTable(G);

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Group action invariants

Low degree resolvents

Subfields

Low degree siblings

Conjugacy classes

Group invariants

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	30T26
Degree	$30$
Order	$120$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_6\times F_5$