LMFDB - Galois group: 45T37

Show commands: Magma

magma:G := TransitiveGroup(45, 37);

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 $t$:	$37$	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:	$(1,44,12,7,21,16,30,26,38,36,3,43,11,8,20,18,29,25,39,35,2,45,10,9,19,17,28,27,37,34)(4,31,14,40,24,5,33,15,42,23)(6,32,13,41,22)$, $(1,3)(4,36)(5,35)(6,34)(7,24)(8,23)(9,22)(10,11)(13,43)(14,45)(15,44)(16,32)(17,31)(18,33)(19,21)(25,41)(26,40)(27,42)(29,30)(37,39)$	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: $S_3$
Degree 5: $C_5$
Degree 9: $C_3^2 : C_6$
Degree 15: $S_3 \times C_5$

Low degree siblings

45T30
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	$2^{20},1^{5}$	$9$	$2$	$20$	$( 2, 3)( 4,35)( 5,34)( 6,36)( 7,22)( 8,24)( 9,23)(10,12)(13,45)(14,44)(15,43)(16,31)(17,33)(18,32)(20,21)(25,40)(26,42)(27,41)(28,29)(37,38)$
3A	$3^{15}$	$2$	$3$	$30$	$( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,11,12)(13,15,14)(16,18,17)(19,20,21)(22,23,24)(25,27,26)(28,29,30)(31,33,32)(34,36,35)(37,39,38)(40,42,41)(43,45,44)$
3B1	$3^{10},1^{15}$	$3$	$3$	$20$	$( 4, 6, 5)( 7, 9, 8)(13,15,14)(16,17,18)(22,23,24)(25,26,27)(31,33,32)(34,35,36)(40,42,41)(43,44,45)$
3B-1	$3^{10},1^{15}$	$3$	$3$	$20$	$( 4, 5, 6)( 7, 8, 9)(13,14,15)(16,18,17)(22,24,23)(25,27,26)(31,32,33)(34,36,35)(40,41,42)(43,45,44)$
3C	$3^{15}$	$6$	$3$	$30$	$( 1,17,33)( 2,16,32)( 3,18,31)( 4,19,35)( 5,21,36)( 6,20,34)( 7,23,37)( 8,24,39)( 9,22,38)(10,27,40)(11,26,42)(12,25,41)(13,28,43)(14,30,44)(15,29,45)$
3D1	$3^{15}$	$6$	$3$	$30$	$( 1,18,33)( 2,17,32)( 3,16,31)( 4,19,36)( 5,21,34)( 6,20,35)( 7,24,39)( 8,22,38)( 9,23,37)(10,25,40)(11,27,42)(12,26,41)(13,28,44)(14,30,45)(15,29,43)$
3D-1	$3^{15}$	$6$	$3$	$30$	$( 1,33,18)( 2,32,17)( 3,31,16)( 4,36,19)( 5,34,21)( 6,35,20)( 7,39,24)( 8,38,22)( 9,37,23)(10,40,25)(11,42,27)(12,41,26)(13,44,28)(14,45,30)(15,43,29)$
5A1	$5^{9}$	$1$	$5$	$36$	$( 1,30,11,39,19)( 2,28,12,38,20)( 3,29,10,37,21)( 4,33,14,42,24)( 5,31,15,40,23)( 6,32,13,41,22)( 7,36,18,45,27)( 8,35,17,44,26)( 9,34,16,43,25)$
5A-1	$5^{9}$	$1$	$5$	$36$	$( 1,19,39,11,30)( 2,20,38,12,28)( 3,21,37,10,29)( 4,24,42,14,33)( 5,23,40,15,31)( 6,22,41,13,32)( 7,27,45,18,36)( 8,26,44,17,35)( 9,25,43,16,34)$
5A2	$5^{9}$	$1$	$5$	$36$	$( 1,11,19,30,39)( 2,12,20,28,38)( 3,10,21,29,37)( 4,14,24,33,42)( 5,15,23,31,40)( 6,13,22,32,41)( 7,18,27,36,45)( 8,17,26,35,44)( 9,16,25,34,43)$
5A-2	$5^{9}$	$1$	$5$	$36$	$( 1,39,30,19,11)( 2,38,28,20,12)( 3,37,29,21,10)( 4,42,33,24,14)( 5,40,31,23,15)( 6,41,32,22,13)( 7,45,36,27,18)( 8,44,35,26,17)( 9,43,34,25,16)$
6A1	$6^{5},2^{5},1^{5}$	$9$	$6$	$30$	$( 2, 3)( 4,34, 6,35, 5,36)( 7,24, 9,22, 8,23)(10,12)(13,44,15,45,14,43)(16,32,17,31,18,33)(20,21)(25,41,26,40,27,42)(28,29)(37,38)$
6A-1	$6^{5},2^{5},1^{5}$	$9$	$6$	$30$	$( 2, 3)( 4,36, 5,35, 6,34)( 7,23, 8,22, 9,24)(10,12)(13,43,14,45,15,44)(16,33,18,31,17,32)(20,21)(25,42,27,40,26,41)(28,29)(37,38)$
10A1	$10^{4},5$	$9$	$10$	$40$	$( 1,39,30,19,11)( 2,37,28,21,12, 3,38,29,20,10)( 4,26,33, 8,14,35,42,17,24,44)( 5,25,31, 9,15,34,40,16,23,43)( 6,27,32, 7,13,36,41,18,22,45)$
10A-1	$10^{4},5$	$9$	$10$	$40$	$( 1,11,19,30,39)( 2,10,20,29,38, 3,12,21,28,37)( 4,44,24,17,42,35,14, 8,33,26)( 5,43,23,16,40,34,15, 9,31,25)( 6,45,22,18,41,36,13, 7,32,27)$
10A3	$10^{4},5$	$9$	$10$	$40$	$( 1,19,39,11,30)( 2,21,38,10,28, 3,20,37,12,29)( 4, 8,42,44,33,35,24,26,14,17)( 5, 9,40,43,31,34,23,25,15,16)( 6, 7,41,45,32,36,22,27,13,18)$
10A-3	$10^{4},5$	$9$	$10$	$40$	$( 1,30,11,39,19)( 2,29,12,37,20, 3,28,10,38,21)( 4,17,14,26,24,35,33,44,42, 8)( 5,16,15,25,23,34,31,43,40, 9)( 6,18,13,27,22,36,32,45,41, 7)$
15A1	$15^{3}$	$2$	$15$	$42$	$( 1,10,20,30,37, 2,11,21,28,39, 3,12,19,29,38)( 4,15,22,33,40, 6,14,23,32,42, 5,13,24,31,41)( 7,16,26,36,43, 8,18,25,35,45, 9,17,27,34,44)$
15A-1	$15^{3}$	$2$	$15$	$42$	$( 1,37,28,19,10, 2,39,29,20,11, 3,38,30,21,12)( 4,40,32,24,15, 6,42,31,22,14, 5,41,33,23,13)( 7,43,35,27,16, 8,45,34,26,18, 9,44,36,25,17)$
15A2	$15^{3}$	$2$	$15$	$42$	$( 1,20,37,11,28, 3,19,38,10,30, 2,21,39,12,29)( 4,22,40,14,32, 5,24,41,15,33, 6,23,42,13,31)( 7,26,43,18,35, 9,27,44,16,36, 8,25,45,17,34)$
15A-2	$15^{3}$	$2$	$15$	$42$	$( 1,28,10,39,20, 3,30,12,37,19, 2,29,11,38,21)( 4,32,15,42,22, 5,33,13,40,24, 6,31,14,41,23)( 7,35,16,45,26, 9,36,17,43,27, 8,34,18,44,25)$
15B1	$15^{2},5^{3}$	$3$	$15$	$40$	$( 1,11,19,30,39)( 2,12,20,28,38)( 3,10,21,29,37)( 4,15,22,33,40, 6,14,23,32,42, 5,13,24,31,41)( 7,17,25,36,44, 9,18,26,34,45, 8,16,27,35,43)$
15B-1	$15^{2},5^{3}$	$3$	$15$	$40$	$( 1,39,30,19,11)( 2,38,28,20,12)( 3,37,29,21,10)( 4,41,31,24,13, 5,42,32,23,14, 6,40,33,22,15)( 7,43,35,27,16, 8,45,34,26,18, 9,44,36,25,17)$
15B2	$15^{2},5^{3}$	$3$	$15$	$40$	$( 1,19,39,11,30)( 2,20,38,12,28)( 3,21,37,10,29)( 4,22,40,14,32, 5,24,41,15,33, 6,23,42,13,31)( 7,25,44,18,34, 8,27,43,17,36, 9,26,45,16,35)$
15B-2	$15^{2},5^{3}$	$3$	$15$	$40$	$( 1,30,11,39,19)( 2,28,12,38,20)( 3,29,10,37,21)( 4,31,13,42,23, 6,33,15,41,24, 5,32,14,40,22)( 7,35,16,45,26, 9,36,17,43,27, 8,34,18,44,25)$
15B4	$15^{2},5^{3}$	$3$	$15$	$40$	$( 1,39,30,19,11)( 2,38,28,20,12)( 3,37,29,21,10)( 4,40,32,24,15, 6,42,31,22,14, 5,41,33,23,13)( 7,44,34,27,17, 9,45,35,25,18, 8,43,36,26,16)$
15B-4	$15^{2},5^{3}$	$3$	$15$	$40$	$( 1,11,19,30,39)( 2,12,20,28,38)( 3,10,21,29,37)( 4,13,23,33,41, 5,14,22,31,42, 6,15,24,32,40)( 7,16,26,36,43, 8,18,25,35,45, 9,17,27,34,44)$
15B7	$15^{2},5^{3}$	$3$	$15$	$40$	$( 1,19,39,11,30)( 2,20,38,12,28)( 3,21,37,10,29)( 4,23,41,14,31, 6,24,40,13,33, 5,22,42,15,32)( 7,26,43,18,35, 9,27,44,16,36, 8,25,45,17,34)$
15B-7	$15^{2},5^{3}$	$3$	$15$	$40$	$( 1,30,11,39,19)( 2,28,12,38,20)( 3,29,10,37,21)( 4,32,15,42,22, 5,33,13,40,24, 6,31,14,41,23)( 7,34,17,45,25, 8,36,16,44,27, 9,35,18,43,26)$
15C1	$15^{3}$	$6$	$15$	$42$	$( 1,14,26,39, 4,17,30,42, 8,19,33,44,11,24,35)( 2,13,25,38, 6,16,28,41, 9,20,32,43,12,22,34)( 3,15,27,37, 5,18,29,40, 7,21,31,45,10,23,36)$
15C-1	$15^{3}$	$6$	$15$	$42$	$( 1, 6, 7,11,13,18,19,22,27,30,32,36,39,41,45)( 2, 5, 8,12,15,17,20,23,26,28,31,35,38,40,44)( 3, 4, 9,10,14,16,21,24,25,29,33,34,37,42,43)$
15C2	$15^{3}$	$6$	$15$	$42$	$( 1,40,34,30,23,16,11, 5,43,39,31,25,19,15, 9)( 2,42,36,28,24,18,12, 4,45,38,33,27,20,14, 7)( 3,41,35,29,22,17,10, 6,44,37,32,26,21,13, 8)$
15C-2	$15^{3}$	$6$	$15$	$42$	$( 1,24,44,19,42,17,39,14,35,11,33, 8,30, 4,26)( 2,22,43,20,41,16,38,13,34,12,32, 9,28, 6,25)( 3,23,45,21,40,18,37,15,36,10,31, 7,29, 5,27)$
15D1	$15^{3}$	$6$	$15$	$42$	$( 1,14,27,39, 4,18,30,42, 7,19,33,45,11,24,36)( 2,13,26,38, 6,17,28,41, 8,20,32,44,12,22,35)( 3,15,25,37, 5,16,29,40, 9,21,31,43,10,23,34)$
15D-1	$15^{3}$	$6$	$15$	$42$	$( 1, 6, 8,11,13,17,19,22,26,30,32,35,39,41,44)( 2, 5, 9,12,15,16,20,23,25,28,31,34,38,40,43)( 3, 4, 7,10,14,18,21,24,27,29,33,36,37,42,45)$
15D2	$15^{3}$	$6$	$15$	$42$	$( 1,40,36,30,23,18,11, 5,45,39,31,27,19,15, 7)( 2,42,35,28,24,17,12, 4,44,38,33,26,20,14, 8)( 3,41,34,29,22,16,10, 6,43,37,32,25,21,13, 9)$
15D-2	$15^{3}$	$6$	$15$	$42$	$( 1,24,45,19,42,18,39,14,36,11,33, 7,30, 4,27)( 2,22,44,20,41,17,38,13,35,12,32, 8,28, 6,26)( 3,23,43,21,40,16,37,15,34,10,31, 9,29, 5,25)$
15D4	$15^{3}$	$6$	$15$	$42$	$( 1, 6, 9,11,13,16,19,22,25,30,32,34,39,41,43)( 2, 5, 7,12,15,18,20,23,27,28,31,36,38,40,45)( 3, 4, 8,10,14,17,21,24,26,29,33,35,37,42,44)$
15D-4	$15^{3}$	$6$	$15$	$42$	$( 1,14,25,39, 4,16,30,42, 9,19,33,43,11,24,34)( 2,13,27,38, 6,18,28,41, 7,20,32,45,12,22,36)( 3,15,26,37, 5,17,29,40, 8,21,31,44,10,23,35)$
15D7	$15^{3}$	$6$	$15$	$42$	$( 1,40,35,30,23,17,11, 5,44,39,31,26,19,15, 8)( 2,42,34,28,24,16,12, 4,43,38,33,25,20,14, 9)( 3,41,36,29,22,18,10, 6,45,37,32,27,21,13, 7)$
15D-7	$15^{3}$	$6$	$15$	$42$	$( 1,24,43,19,42,16,39,14,34,11,33, 9,30, 4,25)( 2,22,45,20,41,18,38,13,36,12,32, 7,28, 6,27)( 3,23,44,21,40,17,37,15,35,10,31, 8,29, 5,26)$
30A1	$30,10,5$	$9$	$30$	$42$	$( 1,30,11,39,19)( 2,29,12,37,20, 3,28,10,38,21)( 4,18,15,26,22,34,33,45,40, 8, 6,16,14,27,23,35,32,43,42, 7, 5,17,13,25,24,36,31,44,41, 9)$
30A-1	$30,10,5$	$9$	$30$	$42$	$( 1,20,39,12,30, 2,19,38,11,28)( 3,21,37,10,29)( 4, 7,41,43,31,35,24,27,13,16, 5, 8,42,45,32,34,23,26,14,18, 6, 9,40,44,33,36,22,25,15,17)$
30A7	$30,10,5$	$9$	$30$	$42$	$( 1,10,19,29,39, 3,11,21,30,37)( 2,12,20,28,38)( 4,43,23,18,41,35,14, 9,31,27, 6,44,24,16,40,36,13, 8,33,25, 5,45,22,17,42,34,15, 7,32,26)$
30A-7	$30,10,5$	$9$	$30$	$42$	$( 1,39,30,19,11)( 2,37,28,21,12, 3,38,29,20,10)( 4,25,32, 8,15,36,42,16,22,44, 5,27,33, 9,13,35,40,18,24,43, 6,26,31, 7,14,34,41,17,23,45)$
30A11	$30,10,5$	$9$	$30$	$42$	$( 1,30,11,39,19)( 2,29,12,37,20, 3,28,10,38,21)( 4,16,13,26,23,36,33,43,41, 8, 5,18,14,25,22,35,31,45,42, 9, 6,17,15,27,24,34,32,44,40, 7)$
30A-11	$30,10,5$	$9$	$30$	$42$	$( 1,20,39,12,30, 2,19,38,11,28)( 3,21,37,10,29)( 4, 8,40,43,32,36,24,26,15,16, 6, 7,42,44,31,34,22,27,14,17, 5, 9,41,45,33,35,23,25,13,18)$
30A13	$30,10,5$	$9$	$30$	$42$	$( 1,39,30,19,11)( 2,37,28,21,12, 3,38,29,20,10)( 4,27,31, 8,13,34,42,18,23,44, 6,25,33, 7,15,35,41,16,24,45, 5,26,32, 9,14,36,40,17,22,43)$
30A-13	$30,10,5$	$9$	$30$	$42$	$( 1,10,19,29,39, 3,11,21,30,37)( 2,12,20,28,38)( 4,44,22,18,40,34,14, 8,32,27, 5,43,24,17,41,36,15, 9,33,26, 6,45,23,16,42,35,13, 7,31,25)$

Malle's constant $a(G)$: $1/20$

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	45T37

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