LMFDB - Galois group: 36T46

Show commands: Magma

magma:G := TransitiveGroup(36, 46);

Group invariants

Abstract group:	$C_9:D_4$	magma:IdentifyGroup(G);
Order:	$72=2^{3} \cdot 3^{2}$	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 $n$ :	$36$	magma:t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$ :	$46$	magma:t, n := TransitiveGroupIdentification(G); t;
Parity:	$-1$	magma:IsEven(G);
Primitive:	no	magma:IsPrimitive(G);
$\card{\Aut(F/K)}$ :	$2$	magma:Order(Centralizer(SymmetricGroup(n), G));
Generators:	$(1,18,2,17)(3,19,4,20)(5,14,6,13)(7,16,8,15)(9,11,10,12)(21,34,22,33)(23,35,24,36)(25,30,26,29)(27,31,28,32)$ , $(1,16)(2,15)(3,14)(4,13)(5,10)(6,9)(7,12)(8,11)(17,36)(18,35)(19,33)(20,34)(21,29)(22,30)(23,32)(24,31)(27,28)$	magma:Generators(G);

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$2$ : $C_2$ x 3
$4$ : $C_2^2$
$6$ : $S_3$
$8$ : $D_{4}$
$12$ : $D_{6}$
$18$ : $D_{9}$
$24$ : $(C_6\times C_2):C_2$
$36$ : $D_{18}$

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$ :	$C_2$ x 3
$4$ :	$C_2^2$
$6$ :	$S_3$
$8$ :	$D_{4}$
$12$ :	$D_{6}$
$18$ :	$D_{9}$
$24$ :	$(C_6\times C_2):C_2$
$36$ :	$D_{18}$

Subfields

Degree 2: $C_2$
Degree 3: $S_3$
Degree 4: $D_{4}$
Degree 6: $D_{6}$
Degree 9: $D_{9}$
Degree 12: $(C_6\times C_2):C_2$
Degree 18: $D_{18}$

Low degree siblings

36T24
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	$1^{36}$	$1$	$1$	$0$	$()$
2A	$2^{18}$	$1$	$2$	$18$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$
2B	$2^{18}$	$2$	$2$	$18$	$( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)$
2C	$2^{17},1^{2}$	$18$	$2$	$17$	$( 1,32)( 2,31)( 3,29)( 4,30)( 5,26)( 6,25)( 7,27)( 8,28)( 9,24)(10,23)(11,22)(12,21)(13,17)(14,18)(15,19)(16,20)(33,34)$
3A	$3^{12}$	$2$	$3$	$24$	$( 1,16,25)( 2,15,26)( 3,13,27)( 4,14,28)( 5,19,31)( 6,20,32)( 7,17,29)( 8,18,30)( 9,23,34)(10,24,33)(11,22,35)(12,21,36)$
4A	$4^{9}$	$18$	$4$	$27$	$( 1,28, 2,27)( 3,25, 4,26)( 5,21, 6,22)( 7,24, 8,23)( 9,17,10,18)(11,19,12,20)(13,16,14,15)(29,33,30,34)(31,36,32,35)$
6A	$6^{6}$	$2$	$6$	$30$	$( 1,26,16, 2,25,15)( 3,28,13, 4,27,14)( 5,32,19, 6,31,20)( 7,30,17, 8,29,18)( 9,33,23,10,34,24)(11,36,22,12,35,21)$
6B1	$6^{6}$	$2$	$6$	$30$	$( 1,28,16, 4,25,14)( 2,27,15, 3,26,13)( 5,30,19, 8,31,18)( 6,29,20, 7,32,17)( 9,35,23,11,34,22)(10,36,24,12,33,21)$
6B-1	$6^{6}$	$2$	$6$	$30$	$( 1,14,25, 4,16,28)( 2,13,26, 3,15,27)( 5,18,31, 8,19,30)( 6,17,32, 7,20,29)( 9,22,34,11,23,35)(10,21,33,12,24,36)$
9A1	$9^{4}$	$2$	$9$	$32$	$( 1,31,24,16, 5,33,25,19,10)( 2,32,23,15, 6,34,26,20, 9)( 3,29,22,13, 7,35,27,17,11)( 4,30,21,14, 8,36,28,18,12)$
9A2	$9^{4}$	$2$	$9$	$32$	$( 1,24, 5,25,10,31,16,33,19)( 2,23, 6,26, 9,32,15,34,20)( 3,22, 7,27,11,29,13,35,17)( 4,21, 8,28,12,30,14,36,18)$
9A4	$9^{4}$	$2$	$9$	$32$	$( 1, 5,10,16,19,24,25,31,33)( 2, 6, 9,15,20,23,26,32,34)( 3, 7,11,13,17,22,27,29,35)( 4, 8,12,14,18,21,28,30,36)$
18A1	$18^{2}$	$2$	$18$	$34$	$( 1,34,31,26,24,20,16, 9, 5, 2,33,32,25,23,19,15,10, 6)( 3,36,29,28,22,18,13,12, 7, 4,35,30,27,21,17,14,11, 8)$
18A5	$18^{2}$	$2$	$18$	$34$	$( 1,23, 5,26,10,32,16,34,19, 2,24, 6,25, 9,31,15,33,20)( 3,21, 7,28,11,30,13,36,17, 4,22, 8,27,12,29,14,35,18)$
18A7	$18^{2}$	$2$	$18$	$34$	$( 1, 9,19,26,33, 6,16,23,31, 2,10,20,25,34, 5,15,24,32)( 3,12,17,28,35, 8,13,21,29, 4,11,18,27,36, 7,14,22,30)$
18B1	$18^{2}$	$2$	$18$	$34$	$( 1,36,31,28,24,18,16,12, 5, 4,33,30,25,21,19,14,10, 8)( 2,35,32,27,23,17,15,11, 6, 3,34,29,26,22,20,13, 9, 7)$
18B-1	$18^{2}$	$2$	$18$	$34$	$( 1, 8,10,14,19,21,25,30,33, 4, 5,12,16,18,24,28,31,36)( 2, 7, 9,13,20,22,26,29,34, 3, 6,11,15,17,23,27,32,35)$
18B5	$18^{2}$	$2$	$18$	$34$	$( 1,18,33,14,31,12,25, 8,24, 4,19,36,16,30,10,28, 5,21)( 2,17,34,13,32,11,26, 7,23, 3,20,35,15,29, 9,27, 6,22)$
18B-5	$18^{2}$	$2$	$18$	$34$	$( 1,21, 5,28,10,30,16,36,19, 4,24, 8,25,12,31,14,33,18)( 2,22, 6,27, 9,29,15,35,20, 3,23, 7,26,11,32,13,34,17)$
18B7	$18^{2}$	$2$	$18$	$34$	$( 1,29,24,13, 5,35,25,17,10, 3,31,22,16, 7,33,27,19,11)( 2,30,23,14, 6,36,26,18, 9, 4,32,21,15, 8,34,28,20,12)$
18B-7	$18^{2}$	$2$	$18$	$34$	$( 1,11,19,27,33, 7,16,22,31, 3,10,17,25,35, 5,13,24,29)( 2,12,20,28,34, 8,15,21,32, 4, 9,18,26,36, 6,14,23,30)$

Malle's constant $a(G)$ : $1/17$

magma:ConjugacyClasses(G);

Character table

		1A	2A	2B	2C	3A	4A	6A	6B1	6B-1	9A1	9A2	9A4	18A1	18A5	18A7	18B1	18B-1	18B5	18B-5	18B7	18B-7
	Size	1	1	2	18	2	18	2	2	2	2	2	2	2	2	2	2	2	2	2	2	2
	2 P	1A	1A	1A	1A	3A	2A	3A	3A	3A	9A2	9A4	9A1	9A1	9A4	9A2	9A1	9A1	9A4	9A4	9A2	9A2
	3 P	1A	2A	2B	2C	1A	4A	2A	2B	2B	3A	3A	3A	6A	6A	6A	6B1	6B-1	6B-1	6B1	6B1	6B-1
	Type
72.8.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
72.8.1b	R	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$
72.8.1c	R	$1$	$1$	$- 1$	$1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$
72.8.1d	R	$1$	$1$	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
72.8.2a	R	$2$	$2$	$2$	$0$	$2$	$0$	$2$	$2$	$2$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$
72.8.2b	R	$2$	$- 2$	$0$	$0$	$2$	$0$	$- 2$	$0$	$0$	$2$	$2$	$2$	$0$	$0$	$0$	$- 2$	$- 2$	$0$	$0$	$0$	$- 2$
72.8.2c	R	$2$	$2$	$- 2$	$0$	$2$	$0$	$2$	$- 2$	$- 2$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$
72.8.2d1	C	$2$	$- 2$	$0$	$0$	$2$	$0$	$- 2$	$0$	$0$	$- 1$	$- 1$	$- 1$	$- 1 - 2 ζ_{3}$	$1 + 2 ζ_{3}$	$1 + 2 ζ_{3}$	$1$	$1$	$- 1 - 2 ζ_{3}$	$- 1 - 2 ζ_{3}$	$1 + 2 ζ_{3}$	$1$
72.8.2d2	C	$2$	$- 2$	$0$	$0$	$2$	$0$	$- 2$	$0$	$0$	$- 1$	$- 1$	$- 1$	$1 + 2 ζ_{3}$	$- 1 - 2 ζ_{3}$	$- 1 - 2 ζ_{3}$	$1$	$1$	$1 + 2 ζ_{3}$	$1 + 2 ζ_{3}$	$- 1 - 2 ζ_{3}$	$1$
72.8.2e1	R	$2$	$2$	$2$	$0$	$- 1$	$0$	$- 1$	$- 1$	$- 1$	$ζ_{9}^{- 4} + ζ_{9}^{4}$	$ζ_{9}^{- 1} + ζ_{9}$	$ζ_{9}^{- 2} + ζ_{9}^{2}$	$ζ_{9}^{- 2} + ζ_{9}^{2}$	$ζ_{9}^{- 1} + ζ_{9}$	$ζ_{9}^{- 2} + ζ_{9}^{2}$	$ζ_{9}^{- 2} + ζ_{9}^{2}$	$ζ_{9}^{- 4} + ζ_{9}^{4}$	$ζ_{9}^{- 4} + ζ_{9}^{4}$	$ζ_{9}^{- 1} + ζ_{9}$	$ζ_{9}^{- 4} + ζ_{9}^{4}$	$ζ_{9}^{- 1} + ζ_{9}$
72.8.2e2	R	$2$	$2$	$2$	$0$	$- 1$	$0$	$- 1$	$- 1$	$- 1$	$ζ_{9}^{- 2} + ζ_{9}^{2}$	$ζ_{9}^{- 4} + ζ_{9}^{4}$	$ζ_{9}^{- 1} + ζ_{9}$	$ζ_{9}^{- 1} + ζ_{9}$	$ζ_{9}^{- 4} + ζ_{9}^{4}$	$ζ_{9}^{- 1} + ζ_{9}$	$ζ_{9}^{- 1} + ζ_{9}$	$ζ_{9}^{- 2} + ζ_{9}^{2}$	$ζ_{9}^{- 2} + ζ_{9}^{2}$	$ζ_{9}^{- 4} + ζ_{9}^{4}$	$ζ_{9}^{- 2} + ζ_{9}^{2}$	$ζ_{9}^{- 4} + ζ_{9}^{4}$
72.8.2e3	R	$2$	$2$	$2$	$0$	$- 1$	$0$	$- 1$	$- 1$	$- 1$	$ζ_{9}^{- 1} + ζ_{9}$	$ζ_{9}^{- 2} + ζ_{9}^{2}$	$ζ_{9}^{- 4} + ζ_{9}^{4}$	$ζ_{9}^{- 4} + ζ_{9}^{4}$	$ζ_{9}^{- 2} + ζ_{9}^{2}$	$ζ_{9}^{- 4} + ζ_{9}^{4}$	$ζ_{9}^{- 4} + ζ_{9}^{4}$	$ζ_{9}^{- 1} + ζ_{9}$	$ζ_{9}^{- 1} + ζ_{9}$	$ζ_{9}^{- 2} + ζ_{9}^{2}$	$ζ_{9}^{- 1} + ζ_{9}$	$ζ_{9}^{- 2} + ζ_{9}^{2}$
72.8.2f1	R	$2$	$2$	$- 2$	$0$	$- 1$	$0$	$- 1$	$1$	$1$	$ζ_{9}^{- 4} + ζ_{9}^{4}$	$ζ_{9}^{- 1} + ζ_{9}$	$ζ_{9}^{- 2} + ζ_{9}^{2}$	$- ζ_{9}^{- 2} - ζ_{9}^{2}$	$- ζ_{9}^{- 1} - ζ_{9}$	$- ζ_{9}^{- 2} - ζ_{9}^{2}$	$ζ_{9}^{- 2} + ζ_{9}^{2}$	$ζ_{9}^{- 4} + ζ_{9}^{4}$	$- ζ_{9}^{- 4} - ζ_{9}^{4}$	$- ζ_{9}^{- 1} - ζ_{9}$	$- ζ_{9}^{- 4} - ζ_{9}^{4}$	$ζ_{9}^{- 1} + ζ_{9}$
72.8.2f2	R	$2$	$2$	$- 2$	$0$	$- 1$	$0$	$- 1$	$1$	$1$	$ζ_{9}^{- 2} + ζ_{9}^{2}$	$ζ_{9}^{- 4} + ζ_{9}^{4}$	$ζ_{9}^{- 1} + ζ_{9}$	$- ζ_{9}^{- 1} - ζ_{9}$	$- ζ_{9}^{- 4} - ζ_{9}^{4}$	$- ζ_{9}^{- 1} - ζ_{9}$	$ζ_{9}^{- 1} + ζ_{9}$	$ζ_{9}^{- 2} + ζ_{9}^{2}$	$- ζ_{9}^{- 2} - ζ_{9}^{2}$	$- ζ_{9}^{- 4} - ζ_{9}^{4}$	$- ζ_{9}^{- 2} - ζ_{9}^{2}$	$ζ_{9}^{- 4} + ζ_{9}^{4}$
72.8.2f3	R	$2$	$2$	$- 2$	$0$	$- 1$	$0$	$- 1$	$1$	$1$	$ζ_{9}^{- 1} + ζ_{9}$	$ζ_{9}^{- 2} + ζ_{9}^{2}$	$ζ_{9}^{- 4} + ζ_{9}^{4}$	$- ζ_{9}^{- 4} - ζ_{9}^{4}$	$- ζ_{9}^{- 2} - ζ_{9}^{2}$	$- ζ_{9}^{- 4} - ζ_{9}^{4}$	$ζ_{9}^{- 4} + ζ_{9}^{4}$	$ζ_{9}^{- 1} + ζ_{9}$	$- ζ_{9}^{- 1} - ζ_{9}$	$- ζ_{9}^{- 2} - ζ_{9}^{2}$	$- ζ_{9}^{- 1} - ζ_{9}$	$ζ_{9}^{- 2} + ζ_{9}^{2}$
72.8.2g1	C	$2$	$- 2$	$0$	$0$	$- 1$	$0$	$1$	$- 1 - 2 ζ_{9}^{3}$	$1 + 2 ζ_{9}^{3}$	$ζ_{9}^{- 4} + ζ_{9}^{4}$	$ζ_{9}^{- 1} + ζ_{9}$	$ζ_{9}^{- 2} + ζ_{9}^{2}$	$- ζ_{9} - ζ_{9}^{2} - ζ_{9}^{4}$	$- ζ_{9}^{- 4} - ζ_{9} - ζ_{9}^{2}$	$ζ_{9} + ζ_{9}^{2} + ζ_{9}^{4}$	$- ζ_{9}^{- 2} - ζ_{9}^{2}$	$- ζ_{9}^{- 4} - ζ_{9}^{4}$	$- ζ_{9}^{- 4} + ζ_{9}^{4}$	$ζ_{9}^{- 4} + ζ_{9} + ζ_{9}^{2}$	$ζ_{9}^{- 4} - ζ_{9}^{4}$	$- ζ_{9}^{- 1} - ζ_{9}$
72.8.2g2	C	$2$	$- 2$	$0$	$0$	$- 1$	$0$	$1$	$1 + 2 ζ_{9}^{3}$	$- 1 - 2 ζ_{9}^{3}$	$ζ_{9}^{- 4} + ζ_{9}^{4}$	$ζ_{9}^{- 1} + ζ_{9}$	$ζ_{9}^{- 2} + ζ_{9}^{2}$	$ζ_{9} + ζ_{9}^{2} + ζ_{9}^{4}$	$ζ_{9}^{- 4} + ζ_{9} + ζ_{9}^{2}$	$- ζ_{9} - ζ_{9}^{2} - ζ_{9}^{4}$	$- ζ_{9}^{- 2} - ζ_{9}^{2}$	$- ζ_{9}^{- 4} - ζ_{9}^{4}$	$ζ_{9}^{- 4} - ζ_{9}^{4}$	$- ζ_{9}^{- 4} - ζ_{9} - ζ_{9}^{2}$	$- ζ_{9}^{- 4} + ζ_{9}^{4}$	$- ζ_{9}^{- 1} - ζ_{9}$
72.8.2g3	C	$2$	$- 2$	$0$	$0$	$- 1$	$0$	$1$	$- 1 - 2 ζ_{9}^{3}$	$1 + 2 ζ_{9}^{3}$	$ζ_{9}^{- 2} + ζ_{9}^{2}$	$ζ_{9}^{- 4} + ζ_{9}^{4}$	$ζ_{9}^{- 1} + ζ_{9}$	$ζ_{9}^{- 4} + ζ_{9} + ζ_{9}^{2}$	$ζ_{9}^{- 4} - ζ_{9}^{4}$	$- ζ_{9}^{- 4} - ζ_{9} - ζ_{9}^{2}$	$- ζ_{9}^{- 1} - ζ_{9}$	$- ζ_{9}^{- 2} - ζ_{9}^{2}$	$- ζ_{9} - ζ_{9}^{2} - ζ_{9}^{4}$	$- ζ_{9}^{- 4} + ζ_{9}^{4}$	$ζ_{9} + ζ_{9}^{2} + ζ_{9}^{4}$	$- ζ_{9}^{- 4} - ζ_{9}^{4}$
72.8.2g4	C	$2$	$- 2$	$0$	$0$	$- 1$	$0$	$1$	$1 + 2 ζ_{9}^{3}$	$- 1 - 2 ζ_{9}^{3}$	$ζ_{9}^{- 2} + ζ_{9}^{2}$	$ζ_{9}^{- 4} + ζ_{9}^{4}$	$ζ_{9}^{- 1} + ζ_{9}$	$- ζ_{9}^{- 4} - ζ_{9} - ζ_{9}^{2}$	$- ζ_{9}^{- 4} + ζ_{9}^{4}$	$ζ_{9}^{- 4} + ζ_{9} + ζ_{9}^{2}$	$- ζ_{9}^{- 1} - ζ_{9}$	$- ζ_{9}^{- 2} - ζ_{9}^{2}$	$ζ_{9} + ζ_{9}^{2} + ζ_{9}^{4}$	$ζ_{9}^{- 4} - ζ_{9}^{4}$	$- ζ_{9} - ζ_{9}^{2} - ζ_{9}^{4}$	$- ζ_{9}^{- 4} - ζ_{9}^{4}$
72.8.2g5	C	$2$	$- 2$	$0$	$0$	$- 1$	$0$	$1$	$- 1 - 2 ζ_{9}^{3}$	$1 + 2 ζ_{9}^{3}$	$ζ_{9}^{- 1} + ζ_{9}$	$ζ_{9}^{- 2} + ζ_{9}^{2}$	$ζ_{9}^{- 4} + ζ_{9}^{4}$	$- ζ_{9}^{- 4} + ζ_{9}^{4}$	$ζ_{9} + ζ_{9}^{2} + ζ_{9}^{4}$	$ζ_{9}^{- 4} - ζ_{9}^{4}$	$- ζ_{9}^{- 4} - ζ_{9}^{4}$	$- ζ_{9}^{- 1} - ζ_{9}$	$ζ_{9}^{- 4} + ζ_{9} + ζ_{9}^{2}$	$- ζ_{9} - ζ_{9}^{2} - ζ_{9}^{4}$	$- ζ_{9}^{- 4} - ζ_{9} - ζ_{9}^{2}$	$- ζ_{9}^{- 2} - ζ_{9}^{2}$
72.8.2g6	C	$2$	$- 2$	$0$	$0$	$- 1$	$0$	$1$	$1 + 2 ζ_{9}^{3}$	$- 1 - 2 ζ_{9}^{3}$	$ζ_{9}^{- 1} + ζ_{9}$	$ζ_{9}^{- 2} + ζ_{9}^{2}$	$ζ_{9}^{- 4} + ζ_{9}^{4}$	$ζ_{9}^{- 4} - ζ_{9}^{4}$	$- ζ_{9} - ζ_{9}^{2} - ζ_{9}^{4}$	$- ζ_{9}^{- 4} + ζ_{9}^{4}$	$- ζ_{9}^{- 4} - ζ_{9}^{4}$	$- ζ_{9}^{- 1} - ζ_{9}$	$- ζ_{9}^{- 4} - ζ_{9} - ζ_{9}^{2}$	$ζ_{9} + ζ_{9}^{2} + ζ_{9}^{4}$	$ζ_{9}^{- 4} + ζ_{9} + ζ_{9}^{2}$	$- ζ_{9}^{- 2} - ζ_{9}^{2}$

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	36T46

Degree	$36$
Order	$72$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$ -group	no
Group:	$C_9:D_4$