LMFDB - Galois group: 30T44

Show commands: Magma

magma: G := TransitiveGroup(30, 44);

Group action invariants

Degree $n$:	$30$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$44$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$C_{15}:D_6$
Parity:	$-1$	magma: IsEven(G);
Primitive:	no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:	$3$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:	(1,6)(2,4)(3,5)(7,13)(8,14)(9,15)(16,18,17)(19,30,20,28,21,29)(22,26,23,27,24,25), (1,18)(2,16)(3,17)(4,30)(5,28)(6,29)(7,26)(8,27)(9,25)(10,22)(11,23)(12,24)(13,20)(14,21)(15,19)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$ x 3
$3$: $C_3$
$4$: $C_2^2$
$6$: $S_3$, $C_6$ x 3
$10$: $D_{5}$
$12$: $D_{6}$, $C_6\times C_2$
$18$: $S_3\times C_3$
$20$: $D_{10}$
$30$: $D_5\times C_3$
$36$: $C_6\times S_3$
$60$: $D_5\times S_3$, 30T5

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$3$:	$C_3$
$4$:	$C_2^2$
$6$:	$S_3$, $C_6$ x 3
$10$:	$D_{5}$
$12$:	$D_{6}$, $C_6\times C_2$
$18$:	$S_3\times C_3$
$20$:	$D_{10}$
$30$:	$D_5\times C_3$
$36$:	$C_6\times S_3$
$60$:	$D_5\times S_3$, 30T5

Subfields

Degree 2: $C_2$
Degree 3: None
Degree 5: $D_{5}$
Degree 6: $S_3\times C_3$
Degree 10: $D_{10}$
Degree 15: None

Low degree siblings

45T14
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 $	$1$	$1$	$()$
	$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$2$	$3$	$(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)$
	$ 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$2$	$3$	$(16,18,17)(19,21,20)(22,24,23)(25,27,26)(28,30,29)$
	$ 6, 6, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1 $	$10$	$6$	$( 4,14)( 5,15)( 6,13)( 7,12)( 8,10)( 9,11)(16,28,18,30,17,29)(19,26,21,25,20, 27)(22,23,24)$
	$ 6, 6, 3, 2, 2, 2, 2, 2, 2, 1, 1, 1 $	$10$	$6$	$( 4,14)( 5,15)( 6,13)( 7,12)( 8,10)( 9,11)(16,29,17,30,18,28)(19,27,20,25,21, 26)(22,24,23)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1 $	$5$	$2$	$( 4,14)( 5,15)( 6,13)( 7,12)( 8,10)( 9,11)(16,30)(17,28)(18,29)(19,25)(20,26) (21,27)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$1$	$3$	$( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)(28,29,30)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$2$	$3$	$( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,30,29)$
	$ 6, 6, 6, 6, 3, 3 $	$5$	$6$	$( 1, 2, 3)( 4,15, 6,14, 5,13)( 7,10, 9,12, 8,11)(16,28,18,30,17,29) (19,26,21,25,20,27)(22,23,24)$
	$ 6, 6, 6, 6, 3, 3 $	$10$	$6$	$( 1, 2, 3)( 4,15, 6,14, 5,13)( 7,10, 9,12, 8,11)(16,29,17,30,18,28) (19,27,20,25,21,26)(22,24,23)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$1$	$3$	$( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)(28,30,29)$
	$ 6, 6, 6, 6, 3, 3 $	$5$	$6$	$( 1, 3, 2)( 4,13, 5,14, 6,15)( 7,11, 8,12, 9,10)(16,29,17,30,18,28) (19,27,20,25,21,26)(22,24,23)$
	$ 15, 15 $	$2$	$15$	$( 1, 4, 9,12,14, 3, 6, 8,11,13, 2, 5, 7,10,15)(16,19,24,27,28,18,21,23,26,30, 17,20,22,25,29)$
	$ 15, 15 $	$4$	$15$	$( 1, 4, 9,12,14, 3, 6, 8,11,13, 2, 5, 7,10,15)(16,20,23,27,29,17,21,24,25,30, 18,19,22,26,28)$
	$ 15, 5, 5, 5 $	$4$	$15$	$( 1, 4, 9,12,14, 3, 6, 8,11,13, 2, 5, 7,10,15)(16,21,22,27,30)(17,19,23,25,28) (18,20,24,26,29)$
	$ 15, 15 $	$2$	$15$	$( 1, 5, 8,12,15, 2, 6, 9,10,13, 3, 4, 7,11,14)(16,20,23,27,29,17,21,24,25,30, 18,19,22,26,28)$
	$ 15, 5, 5, 5 $	$4$	$15$	$( 1, 5, 8,12,15, 2, 6, 9,10,13, 3, 4, 7,11,14)(16,21,22,27,30)(17,19,23,25,28) (18,20,24,26,29)$
	$ 5, 5, 5, 5, 5, 5 $	$2$	$5$	$( 1, 6, 7,12,13)( 2, 4, 8,10,14)( 3, 5, 9,11,15)(16,21,22,27,30) (17,19,23,25,28)(18,20,24,26,29)$
	$ 5, 5, 5, 5, 5, 5 $	$2$	$5$	$( 1, 7,13, 6,12)( 2, 8,14, 4,10)( 3, 9,15, 5,11)(16,22,30,21,27) (17,23,28,19,25)(18,24,29,20,26)$
	$ 15, 5, 5, 5 $	$4$	$15$	$( 1, 7,13, 6,12)( 2, 8,14, 4,10)( 3, 9,15, 5,11)(16,23,29,21,25,18,22,28,20, 27,17,24,30,19,26)$
	$ 15, 5, 5, 5 $	$4$	$15$	$( 1, 7,13, 6,12)( 2, 8,14, 4,10)( 3, 9,15, 5,11)(16,24,28,21,26,17,22,29,19, 27,18,23,30,20,25)$
	$ 15, 15 $	$2$	$15$	$( 1, 8,15, 6,10, 3, 7,14, 5,12, 2, 9,13, 4,11)(16,23,29,21,25,18,22,28,20,27, 17,24,30,19,26)$
	$ 15, 15 $	$4$	$15$	$( 1, 8,15, 6,10, 3, 7,14, 5,12, 2, 9,13, 4,11)(16,24,28,21,26,17,22,29,19,27, 18,23,30,20,25)$
	$ 15, 15 $	$2$	$15$	$( 1, 9,14, 6,11, 2, 7,15, 4,12, 3, 8,13, 5,10)(16,24,28,21,26,17,22,29,19,27, 18,23,30,20,25)$
	$ 30 $	$6$	$30$	$( 1,16, 4,19, 9,24,12,27,14,28, 3,18, 6,21, 8,23,11,26,13,30, 2,17, 5,20, 7, 22,10,25,15,29)$
	$ 30 $	$6$	$30$	$( 1,16, 5,20, 8,23,12,27,15,29, 2,17, 6,21, 9,24,10,25,13,30, 3,18, 4,19, 7, 22,11,26,14,28)$
	$ 10, 10, 10 $	$6$	$10$	$( 1,16, 6,21, 7,22,12,27,13,30)( 2,17, 4,19, 8,23,10,25,14,28)( 3,18, 5,20, 9, 24,11,26,15,29)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$15$	$2$	$( 1,16)( 2,17)( 3,18)( 4,28)( 5,29)( 6,30)( 7,27)( 8,25)( 9,26)(10,23)(11,24) (12,22)(13,21)(14,19)(15,20)$
	$ 6, 6, 6, 6, 6 $	$15$	$6$	$( 1,16, 2,17, 3,18)( 4,28, 5,29, 6,30)( 7,27, 8,25, 9,26)(10,23,11,24,12,22) (13,21,14,19,15,20)$
	$ 6, 6, 6, 6, 6 $	$15$	$6$	$( 1,16, 3,18, 2,17)( 4,28, 6,30, 5,29)( 7,27, 9,26, 8,25)(10,23,12,22,11,24) (13,21,15,20,14,19)$
	$ 30 $	$6$	$30$	$( 1,19,10,29, 5,22,13,17, 8,26, 3,21,12,28, 4,24,15,16, 7,25, 2,20,11,30, 6, 23,14,18, 9,27)$
	$ 30 $	$6$	$30$	$( 1,19,11,30, 4,24,13,17, 9,27, 2,20,12,28, 5,22,14,18, 7,25, 3,21,10,29, 6, 23,15,16, 8,26)$
	$ 10, 10, 10 $	$6$	$10$	$( 1,19,12,28, 6,23,13,17, 7,25)( 2,20,10,29, 4,24,14,18, 8,26)( 3,21,11,30, 5, 22,15,16, 9,27)$
	$ 6, 6, 6, 6, 6 $	$3$	$6$	$( 1,22, 2,23, 3,24)( 4,25, 5,26, 6,27)( 7,30, 8,28, 9,29)(10,17,11,18,12,16) (13,21,14,19,15,20)$
	$ 6, 6, 6, 6, 6 $	$3$	$6$	$( 1,22, 3,24, 2,23)( 4,25, 6,27, 5,26)( 7,30, 9,29, 8,28)(10,17,12,16,11,18) (13,21,15,20,14,19)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$3$	$2$	$( 1,22)( 2,23)( 3,24)( 4,25)( 5,26)( 6,27)( 7,30)( 8,28)( 9,29)(10,17)(11,18) (12,16)(13,21)(14,19)(15,20)$

magma: ConjugacyClasses(G);

Group invariants

Order:	$180=2^{2} \cdot 3^{2} \cdot 5$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	not nilpotent
Label:	180.26	magma: IdentifyGroup(G);
Character table:	36 x 36 character table

magma: CharacterTable(G);

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Group action invariants

Low degree resolvents

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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	30T44
Degree	$30$
Order	$180$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_{15}:D_6$