LMFDB - Galois group: 30T41

Show commands: Magma

magma: G := TransitiveGroup(30, 41);

Group action invariants

Degree $n$:	$30$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$41$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$C_5\times S_3^2$
Parity:	$-1$	magma: IsEven(G);
Primitive:	no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:	$5$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:	(2,3)(5,6)(8,9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(29,30), (1,3,2)(4,6,5)(7,9,8)(10,12,11)(13,15,14), (1,26,7,17,13,23,4,29,10,20)(2,27,8,18,14,24,5,30,11,21)(3,25,9,16,15,22,6,28,12,19)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$ x 3
$4$: $C_2^2$
$5$: $C_5$
$6$: $S_3$ x 2
$10$: $C_{10}$ x 3
$12$: $D_{6}$ x 2
$20$: 20T3
$30$: $S_3 \times C_5$ x 2
$36$: $S_3^2$
$60$: 30T12 x 2

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$5$:	$C_5$
$6$:	$S_3$ x 2
$10$:	$C_{10}$ x 3
$12$:	$D_{6}$ x 2
$20$:	20T3
$30$:	$S_3 \times C_5$ x 2
$36$:	$S_3^2$
$60$:	30T12 x 2

Subfields

Degree 2: $C_2$
Degree 3: None
Degree 5: $C_5$
Degree 6: $S_3^2$
Degree 10: $C_{10}$
Degree 15: None

Low degree siblings

45T15
Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

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:	180.28	magma: IdentifyGroup(G);

Character table: not available.

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	30T41
Degree	$30$
Order	$180$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_5\times S_3^2$