LMFDB - Galois group: 35T16

Show commands: Magma

magma: G := TransitiveGroup(35, 16);

Group action invariants

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

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$
$3$: $C_3$
$4$: $C_4$
$6$: $C_6$
$12$: $C_{12}$
$20$: $F_5$
$42$: $F_7$
$60$: $F_5\times C_3$
$84$: 28T12

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$4$:	$C_4$
$6$:	$C_6$
$12$:	$C_{12}$
$20$:	$F_5$
$42$:	$F_7$
$60$:	$F_5\times C_3$
$84$:	28T12

Subfields

Degree 5: $F_5$
Degree 7: $F_7$

Low degree siblings

There are no siblings with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Label	Cycle Type	Size	Order	Representative
	$1^{35}$	$1$	$1$	$()$
	$3^{10},1^{5}$	$7$	$3$	$( 6,11,21)( 7,12,22)( 8,13,23)( 9,14,24)(10,15,25)(16,31,26)(17,32,27) (18,33,28)(19,34,29)(20,35,30)$
	$3^{10},1^{5}$	$7$	$3$	$( 6,21,11)( 7,22,12)( 8,23,13)( 9,24,14)(10,25,15)(16,26,31)(17,27,32) (18,28,33)(19,29,34)(20,30,35)$
	$12^{2},6,4,1$	$35$	$12$	$( 2, 3, 5, 4)( 6,16,11,31,21,26)( 7,18,15,34,22,28,10,19,12,33,25,29) ( 8,20,14,32,23,30, 9,17,13,35,24,27)$
	$12^{2},6,4,1$	$35$	$12$	$( 2, 3, 5, 4)( 6,26,21,31,11,16)( 7,28,25,34,12,18,10,29,22,33,15,19) ( 8,30,24,32,13,20, 9,27,23,35,14,17)$
	$4^{7},2^{3},1$	$35$	$4$	$( 2, 3, 5, 4)( 6,31)( 7,33,10,34)( 8,35, 9,32)(11,26)(12,28,15,29) (13,30,14,27)(16,21)(17,23,20,24)(18,25,19,22)$
	$12^{2},6,4,1$	$35$	$12$	$( 2, 4, 5, 3)( 6,16,11,31,21,26)( 7,19,15,33,22,29,10,18,12,34,25,28) ( 8,17,14,35,23,27, 9,20,13,32,24,30)$
	$12^{2},6,4,1$	$35$	$12$	$( 2, 4, 5, 3)( 6,26,21,31,11,16)( 7,29,25,33,12,19,10,28,22,34,15,18) ( 8,27,24,35,13,17, 9,30,23,32,14,20)$
	$4^{7},2^{3},1$	$35$	$4$	$( 2, 4, 5, 3)( 6,31)( 7,34,10,33)( 8,32, 9,35)(11,26)(12,29,15,28) (13,27,14,30)(16,21)(17,24,20,23)(18,22,19,25)$
	$2^{14},1^{7}$	$5$	$2$	$( 2, 5)( 3, 4)( 7,10)( 8, 9)(12,15)(13,14)(17,20)(18,19)(22,25)(23,24)(27,30) (28,29)(32,35)(33,34)$
	$6^{4},3^{2},2^{2},1$	$35$	$6$	$( 2, 5)( 3, 4)( 6,11,21)( 7,15,22,10,12,25)( 8,14,23, 9,13,24)(16,31,26) (17,35,27,20,32,30)(18,34,28,19,33,29)$
	$6^{4},3^{2},2^{2},1$	$35$	$6$	$( 2, 5)( 3, 4)( 6,21,11)( 7,25,12,10,22,15)( 8,24,13, 9,23,14)(16,26,31) (17,30,32,20,27,35)(18,29,33,19,28,34)$
	$5^{7}$	$4$	$5$	$( 1, 2, 3, 4, 5)( 6, 7, 8, 9,10)(11,12,13,14,15)(16,17,18,19,20) (21,22,23,24,25)(26,27,28,29,30)(31,32,33,34,35)$
	$15^{2},5$	$28$	$15$	$( 1, 2, 3, 4, 5)( 6,12,23, 9,15,21, 7,13,24,10,11,22, 8,14,25)(16,32,28,19,35, 26,17,33,29,20,31,27,18,34,30)$
	$15^{2},5$	$28$	$15$	$( 1, 2, 3, 4, 5)( 6,22,13, 9,25,11, 7,23,14,10,21,12, 8,24,15)(16,27,33,19,30, 31,17,28,34,20,26,32,18,29,35)$
	$7^{5}$	$6$	$7$	$( 1, 6,11,16,21,26,31)( 2, 7,12,17,22,27,32)( 3, 8,13,18,23,28,33) ( 4, 9,14,19,24,29,34)( 5,10,15,20,25,30,35)$
	$14^{2},7$	$30$	$14$	$( 1, 6,11,16,21,26,31)( 2,10,12,20,22,30,32, 5, 7,15,17,25,27,35) ( 3, 9,13,19,23,29,33, 4, 8,14,18,24,28,34)$
	$35$	$12$	$35$	$( 1, 7,13,19,25,26,32, 3, 9,15,16,22,28,34, 5, 6,12,18,24,30,31, 2, 8,14,20, 21,27,33, 4,10,11,17,23,29,35)$
	$35$	$12$	$35$	$( 1, 8,15,17,24,26,33, 5, 7,14,16,23,30,32, 4, 6,13,20,22,29,31, 3,10,12,19, 21,28,35, 2, 9,11,18,25,27,34)$

magma: ConjugacyClasses(G);

Group invariants

Order:	$420=2^{2} \cdot 3 \cdot 5 \cdot 7$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	not nilpotent
Label:	420.15	magma: IdentifyGroup(G);
Character table:

		1A	2A	3A1	3A-1	4A1	4A-1	5A	6A1	6A-1	7A	12A1	12A-1	12A5	12A-5	14A	15A1	15A-1	35A1	35A-1
	Size	1	5	7	7	35	35	4	35	35	6	35	35	35	35	30	28	28	12	12
	2 P	1A	1A	3A-1	3A1	2A	2A	5A	3A-1	3A1	7A	6A1	6A-1	6A-1	6A1	7A	15A-1	15A1	35A-1	35A1
	3 P	1A	2A	1A	1A	4A-1	4A1	5A	2A	2A	7A	4A1	4A-1	4A1	4A-1	14A	5A	5A	35A1	35A-1
	5 P	1A	2A	3A-1	3A1	4A1	4A-1	1A	6A-1	6A1	7A	12A5	12A-5	12A1	12A-1	14A	3A1	3A-1	7A	7A
	7 P	1A	2A	3A1	3A-1	4A-1	4A1	5A	6A1	6A-1	1A	12A-5	12A5	12A-1	12A1	2A	15A1	15A-1	5A	5A
	Type
420.15.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
420.15.1b	R	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$
420.15.1c1	C	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$1$
420.15.1c2	C	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$1$
420.15.1d1	C	$1$	$- 1$	$1$	$1$	$- i$	$i$	$1$	$- 1$	$- 1$	$1$	$i$	$- i$	$i$	$- i$	$- 1$	$1$	$1$	$1$	$1$
420.15.1d2	C	$1$	$- 1$	$1$	$1$	$i$	$- i$	$1$	$- 1$	$- 1$	$1$	$- i$	$i$	$- i$	$i$	$- 1$	$1$	$1$	$1$	$1$
420.15.1e1	C	$1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$- 1$	$- 1$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$1$
420.15.1e2	C	$1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$- 1$	$- 1$	$1$	$ζ_{3}$	$ζ_{3}^{- 1}$	$1$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$1$	$ζ_{3}^{- 1}$	$ζ_{3}$	$1$	$1$
420.15.1f1	C	$1$	$- 1$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$	$- ζ_{12}^{3}$	$ζ_{12}^{3}$	$1$	$ζ_{12}^{2}$	$- ζ_{12}^{4}$	$1$	$- ζ_{12}$	$ζ_{12}^{5}$	$- ζ_{12}^{5}$	$ζ_{12}$	$- 1$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$	$1$	$1$
420.15.1f2	C	$1$	$- 1$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$	$ζ_{12}^{3}$	$- ζ_{12}^{3}$	$1$	$- ζ_{12}^{4}$	$ζ_{12}^{2}$	$1$	$ζ_{12}^{5}$	$- ζ_{12}$	$ζ_{12}$	$- ζ_{12}^{5}$	$- 1$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$	$1$	$1$
420.15.1f3	C	$1$	$- 1$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$	$ζ_{12}^{3}$	$- ζ_{12}^{3}$	$1$	$ζ_{12}^{2}$	$- ζ_{12}^{4}$	$1$	$ζ_{12}$	$- ζ_{12}^{5}$	$ζ_{12}^{5}$	$- ζ_{12}$	$- 1$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$	$1$	$1$
420.15.1f4	C	$1$	$- 1$	$ζ_{12}^{4}$	$- ζ_{12}^{2}$	$- ζ_{12}^{3}$	$ζ_{12}^{3}$	$1$	$- ζ_{12}^{4}$	$ζ_{12}^{2}$	$1$	$- ζ_{12}^{5}$	$ζ_{12}$	$- ζ_{12}$	$ζ_{12}^{5}$	$- 1$	$- ζ_{12}^{2}$	$ζ_{12}^{4}$	$1$	$1$
420.15.4a	R	$4$	$0$	$4$	$4$	$0$	$0$	$- 1$	$0$	$0$	$4$	$0$	$0$	$0$	$0$	$0$	$- 1$	$- 1$	$- 1$	$- 1$
420.15.4b1	C	$4$	$0$	$4 ζ_{3}^{- 1}$	$4 ζ_{3}$	$0$	$0$	$- 1$	$0$	$0$	$4$	$0$	$0$	$0$	$0$	$0$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$- 1$	$- 1$
420.15.4b2	C	$4$	$0$	$4 ζ_{3}$	$4 ζ_{3}^{- 1}$	$0$	$0$	$- 1$	$0$	$0$	$4$	$0$	$0$	$0$	$0$	$0$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$- 1$	$- 1$
420.15.6a	R	$6$	$6$	$0$	$0$	$0$	$0$	$6$	$0$	$0$	$- 1$	$0$	$0$	$0$	$0$	$- 1$	$0$	$0$	$- 1$	$- 1$
420.15.6b	S	$6$	$- 6$	$0$	$0$	$0$	$0$	$6$	$0$	$0$	$- 1$	$0$	$0$	$0$	$0$	$1$	$0$	$0$	$- 1$	$- 1$
420.15.12a1	C	$12$	$0$	$0$	$0$	$0$	$0$	$- 3$	$0$	$0$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$ζ_{35}^{- 15} - ζ_{35}^{- 14} + 2 ζ_{35}^{- 13} + 1 - 2 ζ_{35}^{4} + ζ_{35}^{5} + 2 ζ_{35}^{8} - 2 ζ_{35}^{9} + ζ_{35}^{10} - 2 ζ_{35}^{11} - ζ_{35}^{14} + 2 ζ_{35}^{15} - 2 ζ_{35}^{16}$	$- ζ_{35}^{- 15} + ζ_{35}^{- 14} - 2 ζ_{35}^{- 13} + 2 ζ_{35}^{4} - ζ_{35}^{5} - 2 ζ_{35}^{8} + 2 ζ_{35}^{9} - ζ_{35}^{10} + 2 ζ_{35}^{11} + ζ_{35}^{14} - 2 ζ_{35}^{15} + 2 ζ_{35}^{16}$
420.15.12a2	C	$12$	$0$	$0$	$0$	$0$	$0$	$- 3$	$0$	$0$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- ζ_{35}^{- 15} + ζ_{35}^{- 14} - 2 ζ_{35}^{- 13} + 2 ζ_{35}^{4} - ζ_{35}^{5} - 2 ζ_{35}^{8} + 2 ζ_{35}^{9} - ζ_{35}^{10} + 2 ζ_{35}^{11} + ζ_{35}^{14} - 2 ζ_{35}^{15} + 2 ζ_{35}^{16}$	$ζ_{35}^{- 15} - ζ_{35}^{- 14} + 2 ζ_{35}^{- 13} + 1 - 2 ζ_{35}^{4} + ζ_{35}^{5} + 2 ζ_{35}^{8} - 2 ζ_{35}^{9} + ζ_{35}^{10} - 2 ζ_{35}^{11} - ζ_{35}^{14} + 2 ζ_{35}^{15} - 2 ζ_{35}^{16}$

magma: CharacterTable(G);

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Group action invariants

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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	35T16
Degree	$35$
Order	$420$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_{35}:C_{12}$