LMFDB - Galois group: 35T20

Show commands: Magma

magma: G := TransitiveGroup(35, 20);

Group action invariants

Degree $n$:	$35$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$20$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$F_5\times F_7$
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:	(1,4,5,2)(6,34,10,32)(7,31,9,35)(8,33)(11,29,15,27)(12,26,14,30)(13,28)(16,24,20,22)(17,21,19,25)(18,23), (1,22,13,19,35,6,2,23,14,20,31,7,3,24,15,16,32,8,4,25,11,17,33,9,5,21,12,18,34,10)(26,27,28,29,30)	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
$40$: $F_{5}\times C_2$
$42$: $F_7$
$60$: $F_5\times C_3$
$84$: $F_7 \times C_2$
$120$: 30T26
$168$: 28T26

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
$40$:	$F_{5}\times C_2$
$42$:	$F_7$
$60$:	$F_5\times C_3$
$84$:	$F_7 \times C_2$
$120$:	30T26
$168$:	28T26

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

magma: ConjugacyClasses(G);

Group invariants

Order:	$840=2^{3} \cdot 3 \cdot 5 \cdot 7$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	not nilpotent
Label:	840.139	magma: IdentifyGroup(G);
Character table:	35 x 35 character table

magma: CharacterTable(G);

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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	35T20
Degree	$35$
Order	$840$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$F_5\times F_7$