LMFDB - Galois group: 45T30

Show commands: Magma

magma:G := TransitiveGroup(45, 30);

Group invariants

Abstract group:	$C_3^2:C_{30}$	magma:IdentifyGroup(G);
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	magma:NilpotencyClass(G);

Group action invariants

Degree $n$:	$45$	magma:t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$30$	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:	$(1,8,14,19,27,31,39,45,6,11,16,24,30,35,41)(2,9,13,21,26,33,37,44,5,10,17,23,29,34,40)(3,7,15,20,25,32,38,43,4,12,18,22,28,36,42)$, $(1,8,13,19,27,33,39,45,5,11,16,23,30,35,40)(2,7,15,20,26,31,37,43,4,12,17,24,29,36,42,3,9,14,21,25,32,38,44,6,10,18,22,28,34,41)$	magma:Generators(G);

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$2$: $C_2$
$3$: $C_3$
$5$: $C_5$
$6$: $S_3$, $C_6$
$10$: $C_{10}$
$15$: $C_{15}$
$18$: $S_3\times C_3$
$30$: $S_3 \times C_5$, $C_{30}$
$54$: $C_3^2 : C_6$
$90$: 30T15

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$5$:	$C_5$
$6$:	$S_3$, $C_6$
$10$:	$C_{10}$
$15$:	$C_{15}$
$18$:	$S_3\times C_3$
$30$:	$S_3 \times C_5$, $C_{30}$
$54$:	$C_3^2 : C_6$
$90$:	30T15

Subfields

Degree 3: $C_3$
Degree 5: $C_5$
Degree 9: $C_3^2 : S_3 $
Degree 15: $C_{15}$

Low degree siblings

45T37
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	$1^{45}$	$1$	$1$	$0$	$()$
2A	$2^{15},1^{15}$	$9$	$2$	$15$	$( 2, 3)( 5, 6)( 8, 9)(10,12)(13,14)(16,17)(20,21)(23,24)(26,27)(28,29)(31,33)(34,35)(37,38)(40,41)(44,45)$
3A	$3^{15}$	$2$	$3$	$30$	$( 1, 3, 2)( 4, 5, 6)( 7, 9, 8)(10,11,12)(13,14,15)(16,18,17)(19,20,21)(22,23,24)(25,26,27)(28,29,30)(31,32,33)(34,35,36)(37,39,38)(40,41,42)(43,44,45)$
3B1	$3^{15}$	$3$	$3$	$30$	$( 1,32,18)( 2,31,16)( 3,33,17)( 4,36,19)( 5,34,20)( 6,35,21)( 7,39,22)( 8,37,24)( 9,38,23)(10,41,27)(11,42,25)(12,40,26)(13,44,28)(14,45,29)(15,43,30)$
3B-1	$3^{15}$	$3$	$3$	$30$	$( 1,18,32)( 2,16,31)( 3,17,33)( 4,19,36)( 5,20,34)( 6,21,35)( 7,22,39)( 8,24,37)( 9,23,38)(10,27,41)(11,25,42)(12,26,40)(13,28,44)(14,29,45)(15,30,43)$
3C	$3^{10},1^{15}$	$6$	$3$	$20$	$( 1, 2, 3)( 7, 9, 8)(10,12,11)(16,18,17)(19,21,20)(25,26,27)(28,30,29)(34,35,36)(37,38,39)(43,44,45)$
3D1	$3^{15}$	$6$	$3$	$30$	$( 1,31,16)( 2,33,17)( 3,32,18)( 4,36,20)( 5,34,21)( 6,35,19)( 7,38,22)( 8,39,24)( 9,37,23)(10,40,26)(11,41,27)(12,42,25)(13,44,29)(14,45,30)(15,43,28)$
3D-1	$3^{15}$	$6$	$3$	$30$	$( 1,16,31)( 2,17,33)( 3,18,32)( 4,20,36)( 5,21,34)( 6,19,35)( 7,22,38)( 8,24,39)( 9,23,37)(10,26,40)(11,27,41)(12,25,42)(13,29,44)(14,30,45)(15,28,43)$
5A1	$5^{9}$	$1$	$5$	$36$	$( 1,19,39,11,30)( 2,21,37,10,29)( 3,20,38,12,28)( 4,22,42,15,32)( 5,23,40,13,33)( 6,24,41,14,31)( 7,25,43,18,36)( 8,27,45,16,35)( 9,26,44,17,34)$
5A-1	$5^{9}$	$1$	$5$	$36$	$( 1,30,11,39,19)( 2,29,10,37,21)( 3,28,12,38,20)( 4,32,15,42,22)( 5,33,13,40,23)( 6,31,14,41,24)( 7,36,18,43,25)( 8,35,16,45,27)( 9,34,17,44,26)$
5A2	$5^{9}$	$1$	$5$	$36$	$( 1,39,30,19,11)( 2,37,29,21,10)( 3,38,28,20,12)( 4,42,32,22,15)( 5,40,33,23,13)( 6,41,31,24,14)( 7,43,36,25,18)( 8,45,35,27,16)( 9,44,34,26,17)$
5A-2	$5^{9}$	$1$	$5$	$36$	$( 1,11,19,30,39)( 2,10,21,29,37)( 3,12,20,28,38)( 4,15,22,32,42)( 5,13,23,33,40)( 6,14,24,31,41)( 7,18,25,36,43)( 8,16,27,35,45)( 9,17,26,34,44)$
6A1	$6^{5},3^{5}$	$9$	$6$	$35$	$( 1,18,32)( 2,17,31, 3,16,33)( 4,19,36)( 5,21,34, 6,20,35)( 7,22,39)( 8,23,37, 9,24,38)(10,26,41,12,27,40)(11,25,42)(13,29,44,14,28,45)(15,30,43)$
6A-1	$6^{5},3^{5}$	$9$	$6$	$35$	$( 1,32,18)( 2,33,16, 3,31,17)( 4,36,19)( 5,35,20, 6,34,21)( 7,39,22)( 8,38,24, 9,37,23)(10,40,27,12,41,26)(11,42,25)(13,45,28,14,44,29)(15,43,30)$
10A1	$10^{3},5^{3}$	$9$	$10$	$39$	$( 1,11,19,30,39)( 2,12,21,28,37, 3,10,20,29,38)( 4,15,22,32,42)( 5,14,23,31,40, 6,13,24,33,41)( 7,18,25,36,43)( 8,17,27,34,45, 9,16,26,35,44)$
10A-1	$10^{3},5^{3}$	$9$	$10$	$39$	$( 1,39,30,19,11)( 2,38,29,20,10, 3,37,28,21,12)( 4,42,32,22,15)( 5,41,33,24,13, 6,40,31,23,14)( 7,43,36,25,18)( 8,44,35,26,16, 9,45,34,27,17)$
10A3	$10^{3},5^{3}$	$9$	$10$	$39$	$( 1,30,11,39,19)( 2,28,10,38,21, 3,29,12,37,20)( 4,32,15,42,22)( 5,31,13,41,23, 6,33,14,40,24)( 7,36,18,43,25)( 8,34,16,44,27, 9,35,17,45,26)$
10A-3	$10^{3},5^{3}$	$9$	$10$	$39$	$( 1,19,39,11,30)( 2,20,37,12,29, 3,21,38,10,28)( 4,22,42,15,32)( 5,24,40,14,33, 6,23,41,13,31)( 7,25,43,18,36)( 8,26,45,17,35, 9,27,44,16,34)$
15A1	$15^{3}$	$2$	$15$	$42$	$( 1,37,28,19,10, 3,39,29,20,11, 2,38,30,21,12)( 4,41,33,22,14, 5,42,31,23,15, 6,40,32,24,13)( 7,45,34,25,16, 9,43,35,26,18, 8,44,36,27,17)$
15A-1	$15^{3}$	$2$	$15$	$42$	$( 1,10,20,30,37, 3,11,21,28,39, 2,12,19,29,38)( 4,14,23,32,41, 5,15,24,33,42, 6,13,22,31,40)( 7,16,26,36,45, 9,18,27,34,43, 8,17,25,35,44)$
15A2	$15^{3}$	$2$	$15$	$42$	$( 1,28,10,39,20, 2,30,12,37,19, 3,29,11,38,21)( 4,33,14,42,23, 6,32,13,41,22, 5,31,15,40,24)( 7,34,16,43,26, 8,36,17,45,25, 9,35,18,44,27)$
15A-2	$15^{3}$	$2$	$15$	$42$	$( 1,20,37,11,28, 2,19,38,10,30, 3,21,39,12,29)( 4,23,41,15,33, 6,22,40,14,32, 5,24,42,13,31)( 7,26,45,18,34, 8,25,44,16,36, 9,27,43,17,35)$
15B1	$15^{3}$	$3$	$15$	$42$	$( 1, 7,15,19,25,32,39,43, 4,11,18,22,30,36,42)( 2, 8,14,21,27,31,37,45, 6,10,16,24,29,35,41)( 3, 9,13,20,26,33,38,44, 5,12,17,23,28,34,40)$
15B-1	$15^{3}$	$3$	$15$	$42$	$( 1,42,36,30,22,18,11, 4,43,39,32,25,19,15, 7)( 2,41,35,29,24,16,10, 6,45,37,31,27,21,14, 8)( 3,40,34,28,23,17,12, 5,44,38,33,26,20,13, 9)$
15B2	$15^{3}$	$3$	$15$	$42$	$( 1,15,25,39, 4,18,30,42, 7,19,32,43,11,22,36)( 2,14,27,37, 6,16,29,41, 8,21,31,45,10,24,35)( 3,13,26,38, 5,17,28,40, 9,20,33,44,12,23,34)$
15B-2	$15^{3}$	$3$	$15$	$42$	$( 1,36,22,11,43,32,19, 7,42,30,18, 4,39,25,15)( 2,35,24,10,45,31,21, 8,41,29,16, 6,37,27,14)( 3,34,23,12,44,33,20, 9,40,28,17, 5,38,26,13)$
15B4	$15^{3}$	$3$	$15$	$42$	$( 1,25, 4,30, 7,32,11,36,15,39,18,42,19,43,22)( 2,27, 6,29, 8,31,10,35,14,37,16,41,21,45,24)( 3,26, 5,28, 9,33,12,34,13,38,17,40,20,44,23)$
15B-4	$15^{3}$	$3$	$15$	$42$	$( 1,22,43,19,42,18,39,15,36,11,32, 7,30, 4,25)( 2,24,45,21,41,16,37,14,35,10,31, 8,29, 6,27)( 3,23,44,20,40,17,38,13,34,12,33, 9,28, 5,26)$
15B7	$15^{3}$	$3$	$15$	$42$	$( 1,43,42,39,36,32,30,25,22,19,18,15,11, 7, 4)( 2,45,41,37,35,31,29,27,24,21,16,14,10, 8, 6)( 3,44,40,38,34,33,28,26,23,20,17,13,12, 9, 5)$
15B-7	$15^{3}$	$3$	$15$	$42$	$( 1, 4, 7,11,15,18,19,22,25,30,32,36,39,42,43)( 2, 6, 8,10,14,16,21,24,27,29,31,35,37,41,45)( 3, 5, 9,12,13,17,20,23,26,28,33,34,38,40,44)$
15C1	$15^{2},5^{3}$	$6$	$15$	$40$	$( 1,20,37,11,28, 2,19,38,10,30, 3,21,39,12,29)( 4,22,42,15,32)( 5,23,40,13,33)( 6,24,41,14,31)( 7,27,44,18,35, 9,25,45,17,36, 8,26,43,16,34)$
15C-1	$15^{2},5^{3}$	$6$	$15$	$40$	$( 1,29,12,39,21, 3,30,10,38,19, 2,28,11,37,20)( 4,33,14,42,23, 6,32,13,41,22, 5,31,15,40,24)( 7,36,18,43,25)( 8,35,16,45,27)( 9,34,17,44,26)$
15C2	$15^{2},5^{3}$	$6$	$15$	$40$	$( 1,39,30,19,11)( 2,37,29,21,10)( 3,38,28,20,12)( 4,41,33,22,14, 5,42,31,23,15, 6,40,32,24,13)( 7,44,35,25,17, 8,43,34,27,18, 9,45,36,26,16)$
15C-2	$15^{2},5^{3}$	$6$	$15$	$40$	$( 1,12,21,30,38, 2,11,20,29,39, 3,10,19,28,37)( 4,15,22,32,42)( 5,13,23,33,40)( 6,14,24,31,41)( 7,16,26,36,45, 9,18,27,34,43, 8,17,25,35,44)$
15D1	$15^{3}$	$6$	$15$	$42$	$( 1,35,24,11,45,31,19, 8,41,30,16, 6,39,27,14)( 2,34,23,10,44,33,21, 9,40,29,17, 5,37,26,13)( 3,36,22,12,43,32,20, 7,42,28,18, 4,38,25,15)$
15D-1	$15^{3}$	$6$	$15$	$42$	$( 1,13,26,39, 5,17,30,40, 9,19,33,44,11,23,34)( 2,15,25,37, 4,18,29,42, 7,21,32,43,10,22,36)( 3,14,27,38, 6,16,28,41, 8,20,31,45,12,24,35)$
15D2	$15^{3}$	$6$	$15$	$42$	$( 1,24,43,19,41,18,39,14,36,11,31, 7,30, 6,25)( 2,23,45,21,40,16,37,13,35,10,33, 8,29, 5,27)( 3,22,44,20,42,17,38,15,34,12,32, 9,28, 4,26)$
15D-2	$15^{3}$	$6$	$15$	$42$	$( 1,27, 6,30, 8,31,11,35,14,39,16,41,19,45,24)( 2,26, 5,29, 9,33,10,34,13,37,17,40,21,44,23)( 3,25, 4,28, 7,32,12,36,15,38,18,42,20,43,22)$
15D4	$15^{3}$	$6$	$15$	$42$	$( 1,43,40,39,36,33,30,25,23,19,18,13,11, 7, 5)( 2,45,42,37,35,32,29,27,22,21,16,15,10, 8, 4)( 3,44,41,38,34,31,28,26,24,20,17,14,12, 9, 6)$
15D-4	$15^{3}$	$6$	$15$	$42$	$( 1, 4, 8,11,15,16,19,22,27,30,32,35,39,42,45)( 2, 6, 9,10,14,17,21,24,26,29,31,34,37,41,44)( 3, 5, 7,12,13,18,20,23,25,28,33,36,38,40,43)$
15D7	$15^{3}$	$6$	$15$	$42$	$( 1, 9,15,19,26,32,39,44, 4,11,17,22,30,34,42)( 2, 7,14,21,25,31,37,43, 6,10,18,24,29,36,41)( 3, 8,13,20,27,33,38,45, 5,12,16,23,28,35,40)$
15D-7	$15^{3}$	$6$	$15$	$42$	$( 1,42,35,30,22,16,11, 4,45,39,32,27,19,15, 8)( 2,41,34,29,24,17,10, 6,44,37,31,26,21,14, 9)( 3,40,36,28,23,18,12, 5,43,38,33,25,20,13, 7)$
30A1	$30,15$	$9$	$30$	$43$	$( 1, 4, 7,11,15,18,19,22,25,30,32,36,39,42,43)( 2, 5, 8,12,14,17,21,23,27,28,31,34,37,40,45, 3, 6, 9,10,13,16,20,24,26,29,33,35,38,41,44)$
30A-1	$30,15$	$9$	$30$	$43$	$( 1,44,42,38,36,33,30,26,22,20,18,13,11, 9, 4, 3,43,40,39,34,32,28,25,23,19,17,15,12, 7, 5)( 2,45,41,37,35,31,29,27,24,21,16,14,10, 8, 6)$
30A7	$30,15$	$9$	$30$	$43$	$( 1,24,43,21,42,16,39,14,36,10,32, 8,30, 6,25, 2,22,45,19,41,18,37,15,35,11,31, 7,29, 4,27)( 3,23,44,20,40,17,38,13,34,12,33, 9,28, 5,26)$
30A-7	$30,15$	$9$	$30$	$43$	$( 1,25, 4,30, 7,32,11,36,15,39,18,42,19,43,22)( 2,26, 6,28, 8,33,10,34,14,38,16,40,21,44,24, 3,27, 5,29, 9,31,12,35,13,37,17,41,20,45,23)$
30A11	$30,15$	$9$	$30$	$43$	$( 1,36,22,11,43,32,19, 7,42,30,18, 4,39,25,15)( 2,34,24,12,45,33,21, 9,41,28,16, 5,37,26,14, 3,35,23,10,44,31,20, 8,40,29,17, 6,38,27,13)$
30A-11	$30,15$	$9$	$30$	$43$	$( 1,13,25,38, 4,17,30,40, 7,20,32,44,11,23,36, 3,15,26,39, 5,18,28,42, 9,19,33,43,12,22,34)( 2,14,27,37, 6,16,29,41, 8,21,31,45,10,24,35)$
30A13	$30,15$	$9$	$30$	$43$	$( 1,42,36,30,22,18,11, 4,43,39,32,25,19,15, 7)( 2,40,35,28,24,17,10, 5,45,38,31,26,21,13, 8, 3,41,34,29,23,16,12, 6,44,37,33,27,20,14, 9)$
30A-13	$30,15$	$9$	$30$	$43$	$( 1, 8,15,21,25,31,39,45, 4,10,18,24,30,35,42, 2, 7,14,19,27,32,37,43, 6,11,16,22,29,36,41)( 3, 9,13,20,26,33,38,44, 5,12,17,23,28,34,40)$

Malle's constant $a(G)$: $1/15$

magma:ConjugacyClasses(G);

Character table

50 x 50 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	45T30

Degree	$45$
Order	$270$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_3^2:C_{30}$