LMFDB - Galois group: 35T23

Show commands: Magma

magma: G := TransitiveGroup(35, 23);

Group action invariants

Degree $n$:	$35$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$23$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$D_7\times S_5$
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:	(1,18,35,12,28,10,24,5,20,32,13,26,7,22)(2,17,31,15,27,8,21,4,16,34,11,29,6,23,3,19,33,14,30,9,25), (1,2,5)(6,32,7,34,10,35)(8,31)(9,33)(11,27)(12,28,15,26,13,30)(14,29)(16,25)(17,23)(18,24,19,22,20,21)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$ x 3
$4$: $C_2^2$
$14$: $D_{7}$
$28$: $D_{14}$
$120$: $S_5$
$240$: $S_5\times C_2$

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$14$:	$D_{7}$
$28$:	$D_{14}$
$120$:	$S_5$
$240$:	$S_5\times C_2$

Subfields

Degree 5: $S_5$
Degree 7: $D_{7}$

Low degree siblings

42T218
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, 1, 1, 1 $	$1$	$1$	$()$
	$ 7, 7, 7, 7, 7 $	$2$	$7$	$( 1,35,28,24,20,13, 7)( 2,34,30,21,19,15, 6)( 3,31,29,25,16,14, 8) ( 4,33,27,23,17,11, 9)( 5,32,26,22,18,12,10)$
	$ 7, 7, 7, 7, 7 $	$2$	$7$	$( 1,24, 7,28,13,35,20)( 2,21, 6,30,15,34,19)( 3,25, 8,29,14,31,16) ( 4,23, 9,27,11,33,17)( 5,22,10,26,12,32,18)$
	$ 7, 7, 7, 7, 7 $	$2$	$7$	$( 1,28,20, 7,35,24,13)( 2,30,19, 6,34,21,15)( 3,29,16, 8,31,25,14) ( 4,27,17, 9,33,23,11)( 5,26,18,10,32,22,12)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1 $	$7$	$2$	$( 6,34)( 7,35)( 8,31)( 9,33)(10,32)(11,27)(12,26)(13,28)(14,29)(15,30)(16,25) (17,23)(18,22)(19,21)(20,24)$
	$ 14, 7, 7, 7 $	$20$	$14$	$( 1,22, 7,26,13,32,20, 5,24,10,28,12,35,18)( 2,21, 6,30,15,34,19) ( 3,25, 8,29,14,31,16)( 4,23, 9,27,11,33,17)$
	$ 14, 7, 7, 7 $	$20$	$14$	$( 1,10,13,18,24,26,35, 5, 7,12,20,22,28,32)( 2, 6,15,19,21,30,34) ( 3, 8,14,16,25,29,31)( 4, 9,11,17,23,27,33)$
	$ 14, 7, 7, 7 $	$20$	$14$	$( 1,12,24,32, 7,18,28, 5,13,22,35,10,20,26)( 2,15,21,34, 6,19,30) ( 3,14,25,31, 8,16,29)( 4,11,23,33, 9,17,27)$
	$ 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$10$	$2$	$( 1, 5)( 7,10)(12,13)(18,20)(22,24)(26,28)(32,35)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $	$70$	$2$	$( 1,18)( 2,19)( 3,16)( 4,17)( 5,20)( 6,15)( 7,12)( 8,14)( 9,11)(10,13)(21,34) (22,35)(23,33)(24,32)(25,31)(26,28)$
	$ 14, 14, 7 $	$30$	$14$	$( 1,22, 7,26,13,32,20, 5,24,10,28,12,35,18)( 2,23, 6,27,15,33,19, 4,21, 9,30, 11,34,17)( 3,25, 8,29,14,31,16)$
	$ 14, 14, 7 $	$30$	$14$	$( 1,10,13,18,24,26,35, 5, 7,12,20,22,28,32)( 2, 9,15,17,21,27,34, 4, 6,11,19, 23,30,33)( 3, 8,14,16,25,29,31)$
	$ 14, 14, 7 $	$30$	$14$	$( 1,12,24,32, 7,18,28, 5,13,22,35,10,20,26)( 2,11,21,33, 6,17,30, 4,15,23,34, 9,19,27)( 3,14,25,31, 8,16,29)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1 $	$15$	$2$	$( 1, 5)( 2, 4)( 6, 9)( 7,10)(11,15)(12,13)(17,19)(18,20)(21,23)(22,24)(26,28) (27,30)(32,35)(33,34)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1 $	$105$	$2$	$( 1,18)( 2,17)( 3,16)( 4,19)( 5,20)( 6,11)( 7,12)( 8,14)( 9,15)(10,13)(21,33) (22,35)(23,34)(24,32)(25,31)(26,28)(27,30)$
	$ 21, 7, 7 $	$40$	$21$	$( 1,12,21,35,10,19,28, 5,15,24,32, 6,20,26, 2,13,22,34, 7,18,30) ( 3,14,25,31, 8,16,29)( 4,11,23,33, 9,17,27)$
	$ 21, 7, 7 $	$40$	$21$	$( 1,10,15,20,22,30,35, 5, 6,13,18,21,28,32, 2, 7,12,19,24,26,34) ( 3, 8,14,16,25,29,31)( 4, 9,11,17,23,27,33)$
	$ 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$20$	$3$	$( 1, 5, 2)( 6, 7,10)(12,15,13)(18,19,20)(21,24,22)(26,30,28)(32,34,35)$
	$ 21, 7, 7 $	$40$	$21$	$( 1,18,34,13,26, 6,24, 5,19,35,12,30, 7,22, 2,20,32,15,28,10,21) ( 3,16,31,14,29, 8,25)( 4,17,33,11,27, 9,23)$
	$ 6, 6, 6, 3, 2, 2, 2, 2, 2, 2, 1, 1 $	$140$	$6$	$( 1,26, 2,28, 5,30)( 3,29)( 4,27)( 6,24,10,21, 7,22)( 8,25)( 9,23)(11,17) (12,19,13,18,15,20)(14,16)(32,34,35)$
	$ 21, 14 $	$40$	$42$	$( 1,12,21,35,10,19,28, 5,15,24,32, 6,20,26, 2,13,22,34, 7,18,30) ( 3,11,25,33, 8,17,29, 4,14,23,31, 9,16,27)$
	$ 21, 14 $	$40$	$42$	$( 1,10,15,20,22,30,35, 5, 6,13,18,21,28,32, 2, 7,12,19,24,26,34) ( 3, 9,14,17,25,27,31, 4, 8,11,16,23,29,33)$
	$ 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2 $	$20$	$6$	$( 1, 5, 2)( 3, 4)( 6, 7,10)( 8, 9)(11,14)(12,15,13)(16,17)(18,19,20)(21,24,22) (23,25)(26,30,28)(27,29)(31,33)(32,34,35)$
	$ 21, 14 $	$40$	$42$	$( 1,18,34,13,26, 6,24, 5,19,35,12,30, 7,22, 2,20,32,15,28,10,21) ( 3,17,31,11,29, 9,25, 4,16,33,14,27, 8,23)$
	$ 6, 6, 6, 3, 2, 2, 2, 2, 2, 2, 2 $	$140$	$6$	$( 1,26, 2,28, 5,30)( 3,27)( 4,29)( 6,24,10,21, 7,22)( 8,23)( 9,25)(11,16) (12,19,13,18,15,20)(14,17)(31,33)(32,34,35)$
	$ 28, 7 $	$60$	$28$	$( 1,12,21,33, 7,18,30, 4,13,22,34, 9,20,26, 2,11,24,32, 6,17,28, 5,15,23,35, 10,19,27)( 3,14,25,31, 8,16,29)$
	$ 28, 7 $	$60$	$28$	$( 1,10,15,17,24,26,34, 4, 7,12,19,23,28,32, 2, 9,13,18,21,27,35, 5, 6,11,20, 22,30,33)( 3, 8,14,16,25,29,31)$
	$ 4, 4, 4, 4, 4, 4, 4, 1, 1, 1, 1, 1, 1, 1 $	$30$	$4$	$( 1, 5, 2, 4)( 6, 9, 7,10)(11,13,12,15)(17,20,18,19)(21,23,24,22)(26,30,27,28) (32,34,33,35)$
	$ 28, 7 $	$60$	$28$	$( 1,18,34,11,28,10,21, 4,20,32,15,27, 7,22, 2,17,35,12,30, 9,24, 5,19,33,13, 26, 6,23)( 3,16,31,14,29, 8,25)$
	$ 4, 4, 4, 4, 4, 4, 4, 2, 2, 2, 1 $	$210$	$4$	$( 1,26, 2,27)( 3,29)( 4,28, 5,30)( 6,23, 7,22)( 8,25)( 9,24,10,21) (11,20,12,19)(13,18,15,17)(14,16)(32,34,33,35)$
	$ 35 $	$48$	$35$	$( 1,12,21,33, 8,20,26, 2,11,25,35,10,19,27, 3,13,22,34, 9,16,28, 5,15,23,31, 7,18,30, 4,14,24,32, 6,17,29)$
	$ 35 $	$48$	$35$	$( 1,10,15,17,25,28,32, 2, 9,14,20,22,30,33, 3, 7,12,19,23,29,35, 5, 6,11,16, 24,26,34, 4, 8,13,18,21,27,31)$
	$ 5, 5, 5, 5, 5, 5, 5 $	$24$	$5$	$( 1, 5, 2, 4, 3)( 6, 9, 8, 7,10)(11,14,13,12,15)(16,20,18,19,17) (21,23,25,24,22)(26,30,27,29,28)(31,35,32,34,33)$
	$ 35 $	$48$	$35$	$( 1,18,34,11,29, 7,22, 2,17,31,13,26, 6,23, 3,20,32,15,27, 8,24, 5,19,33,14, 28,10,21, 4,16,35,12,30, 9,25)$
	$ 10, 10, 10, 5 $	$168$	$10$	$( 1,26, 2,27, 3,28, 5,30, 4,29)( 6,23, 8,24,10,21, 9,25, 7,22)(11,16,13,18,15, 17,14,20,12,19)(31,35,32,34,33)$

magma: ConjugacyClasses(G);

Group invariants

Order:	$1680=2^{4} \cdot 3 \cdot 5 \cdot 7$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	no	magma: IsSolvable(G);
Nilpotency class:	not nilpotent
Label:	1680.927	magma: IdentifyGroup(G);
Character table:	35 x 35 character table

magma: CharacterTable(G);

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Group action invariants

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Low degree siblings

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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	35T23
Degree	$35$
Order	$1680$
Cyclic	no
Abelian	no
Solvable	no
Primitive	no
$p$-group	no
Group:	$D_7\times S_5$