LMFDB - Galois group: 30T41

Show commands: Magma

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

Group invariants

Abstract group:	$C_5\times S_3^2$	magma:IdentifyGroup(G);
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	magma:NilpotencyClass(G);

Group action invariants

Degree $n$ :	$30$	magma:t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$ :	$41$	magma:t, n := TransitiveGroupIdentification(G); t;
Parity:	$-1$	magma:IsEven(G);
Primitive:	no	magma:IsPrimitive(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

$\card{(G/N)}$ 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 $\leq 47$

$\card{(G/N)}$	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

Label	Cycle Type	Size	Order	Index	Representative
1A	$1^{30}$	$1$	$1$	$0$	$()$
2A	$2^{15}$	$3$	$2$	$15$	$( 1,22)( 2,23)( 3,24)( 4,25)( 5,26)( 6,27)( 7,28)( 8,29)( 9,30)(10,16)(11,17)(12,18)(13,19)(14,20)(15,21)$
2B	$2^{15}$	$3$	$2$	$15$	$( 1,22)( 2,24)( 3,23)( 4,25)( 5,27)( 6,26)( 7,28)( 8,30)( 9,29)(10,16)(11,18)(12,17)(13,19)(14,21)(15,20)$
2C	$2^{10},1^{10}$	$9$	$2$	$10$	$( 1, 2)( 4, 5)( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(25,26)(28,29)$
3A	$3^{10}$	$2$	$3$	$20$	$( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21)(22,23,24)(25,26,27)(28,29,30)$
3B	$3^{10}$	$2$	$3$	$20$	$( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,18,17)(19,21,20)(22,24,23)(25,27,26)(28,30,29)$
3C	$3^{5},1^{15}$	$4$	$3$	$10$	$(16,18,17)(19,21,20)(22,24,23)(25,27,26)(28,30,29)$
5A1	$5^{6}$	$1$	$5$	$24$	$( 1,13,10, 7, 4)( 2,14,11, 8, 5)( 3,15,12, 9, 6)(16,28,25,22,19)(17,29,26,23,20)(18,30,27,24,21)$
5A-1	$5^{6}$	$1$	$5$	$24$	$( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)(16,19,22,25,28)(17,20,23,26,29)(18,21,24,27,30)$
5A2	$5^{6}$	$1$	$5$	$24$	$( 1,10, 4,13, 7)( 2,11, 5,14, 8)( 3,12, 6,15, 9)(16,25,19,28,22)(17,26,20,29,23)(18,27,21,30,24)$
5A-2	$5^{6}$	$1$	$5$	$24$	$( 1, 7,13, 4,10)( 2, 8,14, 5,11)( 3, 9,15, 6,12)(16,22,28,19,25)(17,23,29,20,26)(18,24,30,21,27)$
6A	$6^{5}$	$6$	$6$	$25$	$( 1,24, 2,22, 3,23)( 4,27, 5,25, 6,26)( 7,30, 8,28, 9,29)(10,18,11,16,12,17)(13,21,14,19,15,20)$
6B	$6^{5}$	$6$	$6$	$25$	$( 1,23, 2,22, 3,24)( 4,26, 5,25, 6,27)( 7,29, 8,28, 9,30)(10,17,11,16,12,18)(13,20,14,19,15,21)$
10A1	$10^{3}$	$3$	$10$	$27$	$( 1,25, 7,16,13,22, 4,28,10,19)( 2,26, 8,17,14,23, 5,29,11,20)( 3,27, 9,18,15,24, 6,30,12,21)$
10A-1	$10^{3}$	$3$	$10$	$27$	$( 1,19,10,28, 4,22,13,16, 7,25)( 2,20,11,29, 5,23,14,17, 8,26)( 3,21,12,30, 6,24,15,18, 9,27)$
10A3	$10^{3}$	$3$	$10$	$27$	$( 1,16, 4,19, 7,22,10,25,13,28)( 2,17, 5,20, 8,23,11,26,14,29)( 3,18, 6,21, 9,24,12,27,15,30)$
10A-3	$10^{3}$	$3$	$10$	$27$	$( 1,28,13,25,10,22, 7,19, 4,16)( 2,29,14,26,11,23, 8,20, 5,17)( 3,30,15,27,12,24, 9,21, 6,18)$
10B1	$10^{3}$	$3$	$10$	$27$	$( 1,28,13,25,10,22, 7,19, 4,16)( 2,30,14,27,11,24, 8,21, 5,18)( 3,29,15,26,12,23, 9,20, 6,17)$
10B-1	$10^{3}$	$3$	$10$	$27$	$( 1,16, 4,19, 7,22,10,25,13,28)( 2,18, 5,21, 8,24,11,27,14,30)( 3,17, 6,20, 9,23,12,26,15,29)$
10B3	$10^{3}$	$3$	$10$	$27$	$( 1,25, 7,16,13,22, 4,28,10,19)( 2,27, 8,18,14,24, 5,30,11,21)( 3,26, 9,17,15,23, 6,29,12,20)$
10B-3	$10^{3}$	$3$	$10$	$27$	$( 1,19,10,28, 4,22,13,16, 7,25)( 2,21,11,30, 5,24,14,18, 8,27)( 3,20,12,29, 6,23,15,17, 9,26)$
10C1	$10^{2},5^{2}$	$9$	$10$	$26$	$( 1, 8,13, 5,10, 2, 7,14, 4,11)( 3, 9,15, 6,12)(16,23,28,20,25,17,22,29,19,26)(18,24,30,21,27)$
10C-1	$10^{2},5^{2}$	$9$	$10$	$26$	$( 1,10, 4,13, 7)( 2,12, 5,15, 8, 3,11, 6,14, 9)(16,27,19,30,22,18,25,21,28,24)(17,26,20,29,23)$
10C3	$10^{2},5^{2}$	$9$	$10$	$26$	$( 1, 5, 7,11,13, 2, 4, 8,10,14)( 3, 6, 9,12,15)(16,20,22,26,28,17,19,23,25,29)(18,21,24,27,30)$
10C-3	$10^{2},5^{2}$	$9$	$10$	$26$	$( 1,15,10, 9, 4, 3,13,12, 7, 6)( 2,14,11, 8, 5)(16,28,25,22,19)(17,30,26,24,20,18,29,27,23,21)$
15A1	$15^{2}$	$2$	$15$	$28$	$( 1,15,11, 7, 6, 2,13,12, 8, 4, 3,14,10, 9, 5)(16,30,26,22,21,17,28,27,23,19,18,29,25,24,20)$
15A-1	$15^{2}$	$2$	$15$	$28$	$( 1, 5, 9,10,14, 3, 4, 8,12,13, 2, 6, 7,11,15)(16,20,24,25,29,18,19,23,27,28,17,21,22,26,30)$
15A2	$15^{2}$	$2$	$15$	$28$	$( 1,11, 6,13, 8, 3,10, 5,15, 7, 2,12, 4,14, 9)(16,26,21,28,23,18,25,20,30,22,17,27,19,29,24)$
15A-2	$15^{2}$	$2$	$15$	$28$	$( 1, 9,14, 4,12, 2, 7,15, 5,10, 3, 8,13, 6,11)(16,24,29,19,27,17,22,30,20,25,18,23,28,21,26)$
15B1	$15^{2}$	$2$	$15$	$28$	$( 1,12, 5,13, 9, 2,10, 6,14, 7, 3,11, 4,15, 8)(16,26,21,28,23,18,25,20,30,22,17,27,19,29,24)$
15B-1	$15^{2}$	$2$	$15$	$28$	$( 1, 8,15, 4,11, 3, 7,14, 6,10, 2, 9,13, 5,12)(16,24,29,19,27,17,22,30,20,25,18,23,28,21,26)$
15B2	$15^{2}$	$2$	$15$	$28$	$( 1, 5, 9,10,14, 3, 4, 8,12,13, 2, 6, 7,11,15)(16,21,23,25,30,17,19,24,26,28,18,20,22,27,29)$
15B-2	$15^{2}$	$2$	$15$	$28$	$( 1,15,11, 7, 6, 2,13,12, 8, 4, 3,14,10, 9, 5)(16,29,27,22,20,18,28,26,24,19,17,30,25,23,21)$
15C1	$15,5^{3}$	$4$	$15$	$26$	$( 1,13,10, 7, 4)( 2,14,11, 8, 5)( 3,15,12, 9, 6)(16,29,27,22,20,18,28,26,24,19,17,30,25,23,21)$
15C-1	$15,5^{3}$	$4$	$15$	$26$	$( 1, 4, 7,10,13)( 2, 5, 8,11,14)( 3, 6, 9,12,15)(16,20,24,25,29,18,19,23,27,28,17,21,22,26,30)$
15C2	$15,5^{3}$	$4$	$15$	$26$	$( 1,11, 6,13, 8, 3,10, 5,15, 7, 2,12, 4,14, 9)(16,25,19,28,22)(17,26,20,29,23)(18,27,21,30,24)$
15C-2	$15,5^{3}$	$4$	$15$	$26$	$( 1, 8,15, 4,11, 3, 7,14, 6,10, 2, 9,13, 5,12)(16,22,28,19,25)(17,23,29,20,26)(18,24,30,21,27)$
30A1	$30$	$6$	$30$	$29$	$( 1,29,15,25,11,24, 7,20, 6,16, 2,30,13,26,12,22, 8,21, 4,17, 3,28,14,27,10,23, 9,19, 5,18)$
30A-1	$30$	$6$	$30$	$29$	$( 1,18, 6,20, 8,22,10,27,15,29, 2,16, 4,21, 9,23,11,25,13,30, 3,17, 5,19, 7,24,12,26,14,28)$
30A7	$30$	$6$	$30$	$29$	$( 1,19,12,30, 5,23,13,16, 9,27, 2,20,10,28, 6,24,14,17, 7,25, 3,21,11,29, 4,22,15,18, 8,26)$
30A-7	$30$	$6$	$30$	$29$	$( 1,26, 9,16,14,24, 4,29,12,19, 2,27, 7,17,15,22, 5,30,10,20, 3,25, 8,18,13,23, 6,28,11,21)$
30B1	$30$	$6$	$30$	$29$	$( 1,21,12,28, 5,23,13,18, 9,25, 2,20,10,30, 6,22,14,17, 7,27, 3,19,11,29, 4,24,15,16, 8,26)$
30B-1	$30$	$6$	$30$	$29$	$( 1,27, 9,16,14,23, 4,30,12,19, 2,26, 7,18,15,22, 5,29,10,21, 3,25, 8,17,13,24, 6,28,11,20)$
30B7	$30$	$6$	$30$	$29$	$( 1,17, 5,19, 9,24,10,26,14,28, 3,18, 4,20, 8,22,12,27,13,29, 2,16, 6,21, 7,23,11,25,15,30)$
30B-7	$30$	$6$	$30$	$29$	$( 1,29,14,25,12,24, 7,20, 5,16, 3,30,13,26,11,22, 9,21, 4,17, 2,28,15,27,10,23, 8,19, 6,18)$

Malle's constant $a(G)$ : $1/10$

magma:ConjugacyClasses(G);

Character table

45 x 45 character table

magma:CharacterTable(G);

Regular extensions

Data not computed

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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:	$C_5\times S_3^2$