LMFDB - Galois group: 45T39

Show commands: Magma

magma: G := TransitiveGroup(45, 39);

Group action invariants

Degree $n$:	$45$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$39$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$\He_3:C_{10}$
Parity:	$-1$	magma: IsEven(G);
Primitive:	no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:	$15$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:	(1,41,21,13,38,32,11,5,28,23,2,40,19,15,37,31,12,4,29,24,3,42,20,14,39,33,10,6,30,22)(7,17,26,34,45,9,18,27,35,43,8,16,25,36,44), (4,5,6)(7,8,9)(13,14,15)(16,18,17)(22,24,23)(25,27,26)(31,32,33)(34,36,35)(40,41,42)(43,45,44), (1,2,3)(4,34,5,35,6,36)(7,24,9,23,8,22)(10,11,12)(13,45,14,43,15,44)(16,31,17,32,18,33)(19,20,21)(25,41,26,42,27,40)(28,29,30)(37,39,38)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$
$5$: $C_5$
$6$: $S_3$ x 4
$10$: $C_{10}$
$18$: $C_3^2:C_2$
$30$: $S_3 \times C_5$ x 4
$54$: $(C_3^2:C_3):C_2$
$90$: 45T10

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$
$5$:	$C_5$
$6$:	$S_3$ x 4
$10$:	$C_{10}$
$18$:	$C_3^2:C_2$
$30$:	$S_3 \times C_5$ x 4
$54$:	$(C_3^2:C_3):C_2$
$90$:	45T10

Subfields

Degree 3: $S_3$
Degree 5: $C_5$
Degree 9: $(C_3^2:C_3):C_2$
Degree 15: $S_3 \times C_5$

Low degree siblings

45T39 x 3
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, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$6$	$3$	$( 4, 5, 6)( 7, 8, 9)(13,14,15)(16,18,17)(22,24,23)(25,27,26)(31,32,33) (34,36,35)(40,41,42)(43,45,44)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$9$	$2$	$( 4,34)( 5,35)( 6,36)( 7,22)( 8,23)( 9,24)(13,45)(14,43)(15,44)(16,33)(17,31) (18,32)(25,40)(26,41)(27,42)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$1$	$3$	$( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,11,12)(13,15,14)(16,18,17)(19,20,21) (22,23,24)(25,27,26)(28,29,30)(31,33,32)(34,36,35)(37,39,38)(40,42,41) (43,45,44)$
	$ 6, 6, 6, 6, 6, 3, 3, 3, 3, 3 $	$9$	$6$	$( 1, 2, 3)( 4,34, 5,35, 6,36)( 7,24, 9,23, 8,22)(10,11,12)(13,45,14,43,15,44) (16,31,17,32,18,33)(19,20,21)(25,41,26,42,27,40)(28,29,30)(37,39,38)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$1$	$3$	$( 1, 3, 2)( 4, 5, 6)( 7, 9, 8)(10,12,11)(13,14,15)(16,17,18)(19,21,20) (22,24,23)(25,26,27)(28,30,29)(31,32,33)(34,35,36)(37,38,39)(40,41,42) (43,44,45)$
	$ 6, 6, 6, 6, 6, 3, 3, 3, 3, 3 $	$9$	$6$	$( 1, 3, 2)( 4,34, 6,36, 5,35)( 7,23, 8,24, 9,22)(10,12,11)(13,45,15,44,14,43) (16,32,18,31,17,33)(19,21,20)(25,42,27,41,26,40)(28,30,29)(37,38,39)$
	$ 15, 15, 15 $	$6$	$15$	$( 1, 4, 7,11,14,18,19,22,25,29,32,34,39,40,43)( 2, 6, 8,12,13,17,20,23,27,30, 31,36,38,42,45)( 3, 5, 9,10,15,16,21,24,26,28,33,35,37,41,44)$
	$ 15, 15, 15 $	$6$	$15$	$( 1, 4, 8,11,14,17,19,22,27,29,32,36,39,40,45)( 2, 6, 9,12,13,16,20,23,26,30, 31,35,38,42,44)( 3, 5, 7,10,15,18,21,24,25,28,33,34,37,41,43)$
	$ 15, 15, 15 $	$6$	$15$	$( 1, 4, 9,11,14,16,19,22,26,29,32,35,39,40,44)( 2, 6, 7,12,13,18,20,23,25,30, 31,34,38,42,43)( 3, 5, 8,10,15,17,21,24,27,28,33,36,37,41,45)$
	$ 30, 15 $	$9$	$30$	$( 1, 4,37,41,30,31,19,22,10,15, 2, 6,39,40,28,33,20,23,11,14, 3, 5,38,42,29, 32,21,24,12,13)( 7,27,44,18,36, 9,25,45,16,34, 8,26,43,17,35)$
	$ 30, 15 $	$9$	$30$	$( 1, 4,38,42,28,33,19,22,12,13, 3, 5,39,40,30,31,21,24,11,14, 2, 6,37,41,29, 32,20,23,10,15)( 7,26,45,18,35, 8,25,44,17,34, 9,27,43,16,36)$
	$ 10, 10, 10, 5, 5, 5 $	$9$	$10$	$( 1, 4,39,40,29,32,19,22,11,14)( 2, 6,38,42,30,31,20,23,12,13)( 3, 5,37,41,28, 33,21,24,10,15)( 7,25,43,18,34)( 8,27,45,17,36)( 9,26,44,16,35)$
	$ 15, 15, 15 $	$6$	$15$	$( 1, 7,13,19,25,31,39,43, 6,11,18,23,29,34,42)( 2, 8,15,20,27,33,38,45, 5,12, 17,24,30,36,41)( 3, 9,14,21,26,32,37,44, 4,10,16,22,28,35,40)$
	$ 15, 15, 15 $	$6$	$15$	$( 1, 7,14,19,25,32,39,43, 4,11,18,22,29,34,40)( 2, 8,13,20,27,31,38,45, 6,12, 17,23,30,36,42)( 3, 9,15,21,26,33,37,44, 5,10,16,24,28,35,41)$
	$ 15, 15, 15 $	$6$	$15$	$( 1, 7,15,19,25,33,39,43, 5,11,18,24,29,34,41)( 2, 8,14,20,27,32,38,45, 4,12, 17,22,30,36,40)( 3, 9,13,21,26,31,37,44, 6,10,16,23,28,35,42)$
	$ 10, 10, 10, 5, 5, 5 $	$9$	$10$	$( 1, 7,29,34,11,18,39,43,19,25)( 2, 8,30,36,12,17,38,45,20,27)( 3, 9,28,35,10, 16,37,44,21,26)( 4,40,32,22,14)( 5,41,33,24,15)( 6,42,31,23,13)$
	$ 30, 15 $	$9$	$30$	$( 1, 7,30,36,10,16,39,43,20,27, 3, 9,29,34,12,17,37,44,19,25, 2, 8,28,35,11, 18,38,45,21,26)( 4,41,31,22,15, 6,40,33,23,14, 5,42,32,24,13)$
	$ 30, 15 $	$9$	$30$	$( 1, 7,28,35,12,17,39,43,21,26, 2, 8,29,34,10,16,38,45,19,25, 3, 9,30,36,11, 18,37,44,20,27)( 4,42,33,22,13, 5,40,31,24,14, 6,41,32,23,15)$
	$ 15, 15, 5, 5, 5 $	$6$	$15$	$( 1,10,20,29,37, 2,11,21,30,39, 3,12,19,28,38)( 4,13,24,32,42, 5,14,23,33,40, 6,15,22,31,41)( 7,18,25,34,43)( 8,17,27,36,45)( 9,16,26,35,44)$
	$ 15, 15, 15 $	$1$	$15$	$( 1,10,20,29,37, 2,11,21,30,39, 3,12,19,28,38)( 4,15,23,32,41, 6,14,24,31,40, 5,13,22,33,42)( 7,16,27,34,44, 8,18,26,36,43, 9,17,25,35,45)$
	$ 30, 15 $	$9$	$30$	$( 1,10,20,29,37, 2,11,21,30,39, 3,12,19,28,38)( 4,43,23,17,41,35,14, 7,31,27, 5,44,22,18,42,36,15, 9,32,25, 6,45,24,16,40,34,13, 8,33,26)$
	$ 5, 5, 5, 5, 5, 5, 5, 5, 5 $	$1$	$5$	$( 1,11,19,29,39)( 2,12,20,30,38)( 3,10,21,28,37)( 4,14,22,32,40) ( 5,15,24,33,41)( 6,13,23,31,42)( 7,18,25,34,43)( 8,17,27,36,45) ( 9,16,26,35,44)$
	$ 10, 10, 10, 5, 5, 5 $	$9$	$10$	$( 1,11,19,29,39)( 2,12,20,30,38)( 3,10,21,28,37)( 4,43,22,18,40,34,14, 7,32,25 )( 5,44,24,16,41,35,15, 9,33,26)( 6,45,23,17,42,36,13, 8,31,27)$
	$ 15, 15, 15 $	$1$	$15$	$( 1,12,21,29,38, 3,11,20,28,39, 2,10,19,30,37)( 4,13,24,32,42, 5,14,23,33,40, 6,15,22,31,41)( 7,17,26,34,45, 9,18,27,35,43, 8,16,25,36,44)$
	$ 30, 15 $	$9$	$30$	$( 1,12,21,29,38, 3,11,20,28,39, 2,10,19,30,37)( 4,43,24,16,42,36,14, 7,33,26, 6,45,22,18,41,35,13, 8,32,25, 5,44,23,17,40,34,15, 9,31,27)$
	$ 30, 15 $	$9$	$30$	$( 1,13,12,24,21,32,29,42,38, 5, 3,14,11,23,20,33,28,40,39, 6, 2,15,10,22,19, 31,30,41,37, 4)( 7,35,17,43,26, 8,34,16,45,25, 9,36,18,44,27)$
	$ 30, 15 $	$9$	$30$	$( 1,13,10,22,20,33,29,42,37, 4, 2,15,11,23,21,32,30,41,39, 6, 3,14,12,24,19, 31,28,40,38, 5)( 7,36,16,43,27, 9,34,17,44,25, 8,35,18,45,26)$
	$ 10, 10, 10, 5, 5, 5 $	$9$	$10$	$( 1,13,11,23,19,31,29,42,39, 6)( 2,15,12,24,20,33,30,41,38, 5)( 3,14,10,22,21, 32,28,40,37, 4)( 7,34,18,43,25)( 8,36,17,45,27)( 9,35,16,44,26)$
	$ 15, 15, 15 $	$6$	$15$	$( 1,13,25,39, 6,18,29,42, 7,19,31,43,11,23,34)( 2,15,27,38, 5,17,30,41, 8,20, 33,45,12,24,36)( 3,14,26,37, 4,16,28,40, 9,21,32,44,10,22,35)$
	$ 15, 15, 15 $	$6$	$15$	$( 1,13,26,39, 6,16,29,42, 9,19,31,44,11,23,35)( 2,15,25,38, 5,18,30,41, 7,20, 33,43,12,24,34)( 3,14,27,37, 4,17,28,40, 8,21,32,45,10,22,36)$
	$ 15, 15, 15 $	$6$	$15$	$( 1,13,27,39, 6,17,29,42, 8,19,31,45,11,23,36)( 2,15,26,38, 5,16,30,41, 9,20, 33,44,12,24,35)( 3,14,25,37, 4,18,28,40, 7,21,32,43,10,22,34)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$6$	$3$	$( 1,16,32)( 2,18,31)( 3,17,33)( 4,19,35)( 5,21,36)( 6,20,34)( 7,23,38) ( 8,24,37)( 9,22,39)(10,27,41)(11,26,40)(12,25,42)(13,30,43)(14,29,44) (15,28,45)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$6$	$3$	$( 1,16,33)( 2,18,32)( 3,17,31)( 4,20,34)( 5,19,35)( 6,21,36)( 7,22,38) ( 8,23,37)( 9,24,39)(10,27,42)(11,26,41)(12,25,40)(13,28,45)(14,30,43) (15,29,44)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$6$	$3$	$( 1,16,31)( 2,18,33)( 3,17,32)( 4,21,36)( 5,20,34)( 6,19,35)( 7,24,38) ( 8,22,37)( 9,23,39)(10,27,40)(11,26,42)(12,25,41)(13,29,44)(14,28,45) (15,30,43)$
	$ 5, 5, 5, 5, 5, 5, 5, 5, 5 $	$1$	$5$	$( 1,19,39,11,29)( 2,20,38,12,30)( 3,21,37,10,28)( 4,22,40,14,32) ( 5,24,41,15,33)( 6,23,42,13,31)( 7,25,43,18,34)( 8,27,45,17,36) ( 9,26,44,16,35)$
	$ 15, 15, 5, 5, 5 $	$6$	$15$	$( 1,19,39,11,29)( 2,20,38,12,30)( 3,21,37,10,28)( 4,23,41,14,31, 5,22,42,15, 32, 6,24,40,13,33)( 7,26,45,18,35, 8,25,44,17,34, 9,27,43,16,36)$
	$ 15, 15, 15 $	$1$	$15$	$( 1,20,37,11,30, 3,19,38,10,29, 2,21,39,12,28)( 4,23,41,14,31, 5,22,42,15,32, 6,24,40,13,33)( 7,27,44,18,36, 9,25,45,16,34, 8,26,43,17,35)$
	$ 15, 15, 15 $	$1$	$15$	$( 1,21,38,11,28, 2,19,37,12,29, 3,20,39,10,30)( 4,24,42,14,33, 6,22,41,13,32, 5,23,40,15,31)( 7,26,45,18,35, 8,25,44,17,34, 9,27,43,16,36)$
	$ 15, 15, 15 $	$6$	$15$	$( 1,25, 6,29, 7,31,11,34,13,39,18,42,19,43,23)( 2,27, 5,30, 8,33,12,36,15,38, 17,41,20,45,24)( 3,26, 4,28, 9,32,10,35,14,37,16,40,21,44,22)$
	$ 15, 15, 15 $	$6$	$15$	$( 1,25, 4,29, 7,32,11,34,14,39,18,40,19,43,22)( 2,27, 6,30, 8,31,12,36,13,38, 17,42,20,45,23)( 3,26, 5,28, 9,33,10,35,15,37,16,41,21,44,24)$
	$ 15, 15, 15 $	$6$	$15$	$( 1,25, 5,29, 7,33,11,34,15,39,18,41,19,43,24)( 2,27, 4,30, 8,32,12,36,14,38, 17,40,20,45,22)( 3,26, 6,28, 9,31,10,35,13,37,16,42,21,44,23)$
	$ 15, 15, 5, 5, 5 $	$6$	$15$	$( 1,28,12,39,21, 2,29,10,38,19, 3,30,11,37,20)( 4,31,15,40,23, 5,32,13,41,22, 6,33,14,42,24)( 7,34,18,43,25)( 8,36,17,45,27)( 9,35,16,44,26)$
	$ 15, 15, 15 $	$1$	$15$	$( 1,28,12,39,21, 2,29,10,38,19, 3,30,11,37,20)( 4,33,13,40,24, 6,32,15,42,22, 5,31,14,41,23)( 7,35,17,43,26, 8,34,16,45,25, 9,36,18,44,27)$
	$ 5, 5, 5, 5, 5, 5, 5, 5, 5 $	$1$	$5$	$( 1,29,11,39,19)( 2,30,12,38,20)( 3,28,10,37,21)( 4,32,14,40,22) ( 5,33,15,41,24)( 6,31,13,42,23)( 7,34,18,43,25)( 8,36,17,45,27) ( 9,35,16,44,26)$
	$ 15, 15, 15 $	$1$	$15$	$( 1,30,10,39,20, 3,29,12,37,19, 2,28,11,38,21)( 4,31,15,40,23, 5,32,13,41,22, 6,33,14,42,24)( 7,36,16,43,27, 9,34,17,44,25, 8,35,18,45,26)$
	$ 15, 15, 5, 5, 5 $	$6$	$15$	$( 1,37,30,19,10, 2,39,28,20,11, 3,38,29,21,12)( 4,40,32,22,14)( 5,41,33,24,15) ( 6,42,31,23,13)( 7,45,35,25,17, 9,43,36,26,18, 8,44,34,27,16)$
	$ 15, 15, 15 $	$1$	$15$	$( 1,37,30,19,10, 2,39,28,20,11, 3,38,29,21,12)( 4,41,31,22,15, 6,40,33,23,14, 5,42,32,24,13)( 7,44,36,25,16, 8,43,35,27,18, 9,45,34,26,17)$
	$ 15, 15, 15 $	$1$	$15$	$( 1,38,28,19,12, 3,39,30,21,11, 2,37,29,20,10)( 4,42,33,22,13, 5,40,31,24,14, 6,41,32,23,15)( 7,45,35,25,17, 9,43,36,26,18, 8,44,34,27,16)$
	$ 5, 5, 5, 5, 5, 5, 5, 5, 5 $	$1$	$5$	$( 1,39,29,19,11)( 2,38,30,20,12)( 3,37,28,21,10)( 4,40,32,22,14) ( 5,41,33,24,15)( 6,42,31,23,13)( 7,43,34,25,18)( 8,45,36,27,17) ( 9,44,35,26,16)$

magma: ConjugacyClasses(G);

Group invariants

Order:	$270=2 \cdot 3^{3} \cdot 5$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	not nilpotent
Label:	270.17	magma: IdentifyGroup(G);
Character table:	50 x 50 character table

magma: CharacterTable(G);

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Elliptic curves over $\Q$
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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	45T39
Degree	$45$
Order	$270$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$\He_3:C_{10}$