LMFDB - Galois group: 36T509

Show commands: Magma

magma:G := TransitiveGroup(36, 509);

Group invariants

Abstract group:	$C_3^2:S_3^2$	magma:IdentifyGroup(G);
Order:	$324=2^{2} \cdot 3^{4}$	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 $t$:	$509$	magma:t, n := TransitiveGroupIdentification(G); t;
Parity:	$1$	magma:IsEven(G);
Primitive:	no	magma:IsPrimitive(G);
$\card{\Aut(F/K)}$:	$6$	magma:Order(Centralizer(SymmetricGroup(n), G));
Generators:	$(1,15,4,14,5,18)(2,16,3,13,6,17)(7,8)(9,11)(10,12)(19,25,24,27,22,30)(20,26,23,28,21,29)(31,32)(33,36)(34,35)$, $(1,26,4,28,5,29)(2,25,3,27,6,30)(7,19,12,22,9,24)(8,20,11,21,10,23)(13,34,16,36,17,32)(14,33,15,35,18,31)$, $(1,21,7,25,13,31)(2,22,8,26,14,32)(3,19,11,29,15,36)(4,20,12,30,16,35)(5,23,9,27,17,33)(6,24,10,28,18,34)$	magma:Generators(G);

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$2$: $C_2$ x 3
$4$: $C_2^2$
$6$: $S_3$ x 5
$12$: $D_{6}$ x 5
$18$: $C_3^2:C_2$
$36$: $S_3^2$ x 4, 18T12
$108$: $C_3^2 : D_{6} $, 18T58

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$6$:	$S_3$ x 5
$12$:	$D_{6}$ x 5
$18$:	$C_3^2:C_2$
$36$:	$S_3^2$ x 4, 18T12
$108$:	$C_3^2 : D_{6} $, 18T58

Subfields

Degree 2: $C_2$ x 3
Degree 3: $S_3$
Degree 4: $C_2^2$
Degree 6: $S_3$, $D_{6}$ x 2
Degree 9: None
Degree 12: $D_6$
Degree 18: 18T139

Low degree siblings

18T133, 18T139, 27T117 x 3, 27T127, 36T506, 36T528
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	$2^{18}$	$9$	$2$	$18$	$( 1,30)( 2,29)( 3,28)( 4,27)( 5,25)( 6,26)( 7,35)( 8,36)( 9,31)(10,32)(11,34)(12,33)(13,20)(14,19)(15,24)(16,23)(17,21)(18,22)$
2B	$2^{18}$	$27$	$2$	$18$	$( 1, 6)( 2, 5)( 3, 4)( 7,18)( 8,17)( 9,14)(10,13)(11,16)(12,15)(19,35)(20,36)(21,34)(22,33)(23,32)(24,31)(25,28)(26,27)(29,30)$
2C	$2^{18}$	$27$	$2$	$18$	$( 1,36)( 2,35)( 3,31)( 4,32)( 5,34)( 6,33)( 7,26)( 8,25)( 9,29)(10,30)(11,27)(12,28)(13,24)(14,23)(15,20)(16,19)(17,22)(18,21)$
3A	$3^{12}$	$2$	$3$	$24$	$( 1, 5, 4)( 2, 6, 3)( 7, 9,12)( 8,10,11)(13,17,16)(14,18,15)(19,22,24)(20,21,23)(25,27,30)(26,28,29)(31,33,35)(32,34,36)$
3B	$3^{12}$	$2$	$3$	$24$	$( 1, 4, 5)( 2, 3, 6)( 7,12, 9)( 8,11,10)(13,16,17)(14,15,18)(19,22,24)(20,21,23)(25,27,30)(26,28,29)(31,33,35)(32,34,36)$
3C	$3^{6},1^{18}$	$4$	$3$	$12$	$( 1, 5, 4)( 2, 6, 3)( 7, 9,12)( 8,10,11)(13,17,16)(14,18,15)$
3D	$3^{12}$	$6$	$3$	$24$	$( 1, 9,16)( 2,10,15)( 3, 8,18)( 4, 7,17)( 5,12,13)( 6,11,14)(19,26,34)(20,25,33)(21,27,35)(22,28,36)(23,30,31)(24,29,32)$
3E	$3^{12}$	$6$	$3$	$24$	$( 1,13, 7)( 2,14, 8)( 3,15,11)( 4,16,12)( 5,17, 9)( 6,18,10)(19,36,29)(20,35,30)(21,31,25)(22,32,26)(23,33,27)(24,34,28)$
3F	$3^{12}$	$6$	$3$	$24$	$( 1,12,17)( 2,11,18)( 3,10,14)( 4, 9,13)( 5, 7,16)( 6, 8,15)(19,28,32)(20,27,31)(21,30,33)(22,29,34)(23,25,35)(24,26,36)$
3G	$3^{8},1^{12}$	$6$	$3$	$16$	$( 7,12, 9)( 8,11,10)(13,17,16)(14,18,15)(19,24,22)(20,23,21)(31,33,35)(32,34,36)$
3H	$3^{12}$	$12$	$3$	$24$	$( 1, 9,13)( 2,10,14)( 3, 8,15)( 4, 7,16)( 5,12,17)( 6,11,18)(19,28,36)(20,27,35)(21,30,31)(22,29,32)(23,25,33)(24,26,34)$
3I	$3^{8},1^{12}$	$12$	$3$	$16$	$( 1, 5, 4)( 2, 6, 3)( 7,12, 9)( 8,11,10)(19,24,22)(20,23,21)(25,27,30)(26,28,29)$
3J	$3^{12}$	$12$	$3$	$24$	$( 1, 9,13)( 2,10,14)( 3, 8,15)( 4, 7,16)( 5,12,17)( 6,11,18)(19,29,34)(20,30,33)(21,25,35)(22,26,36)(23,27,31)(24,28,32)$
3K	$3^{12}$	$12$	$3$	$24$	$( 1, 9,13)( 2,10,14)( 3, 8,15)( 4, 7,16)( 5,12,17)( 6,11,18)(19,26,32)(20,25,31)(21,27,33)(22,28,34)(23,30,35)(24,29,36)$
6A	$6^{6}$	$18$	$6$	$30$	$( 1,23, 9,30,16,31)( 2,24,10,29,15,32)( 3,22, 8,28,18,36)( 4,21, 7,27,17,35)( 5,20,12,25,13,33)( 6,19,11,26,14,34)$
6B	$6^{6}$	$18$	$6$	$30$	$( 1,31,13,25, 7,21)( 2,32,14,26, 8,22)( 3,36,15,29,11,19)( 4,35,16,30,12,20)( 5,33,17,27, 9,23)( 6,34,18,28,10,24)$
6C	$6^{6}$	$18$	$6$	$30$	$( 1,25, 5,27, 4,30)( 2,26, 6,28, 3,29)( 7,31, 9,33,12,35)( 8,32,10,34,11,36)(13,21,17,23,16,20)(14,22,18,24,15,19)$
6D	$6^{6}$	$18$	$6$	$30$	$( 1,21,12,30,17,33)( 2,22,11,29,18,34)( 3,19,10,28,14,32)( 4,20, 9,27,13,31)( 5,23, 7,25,16,35)( 6,24, 8,26,15,36)$
6E	$6^{4},2^{6}$	$54$	$6$	$26$	$( 1, 6)( 2, 5)( 3, 4)( 7,14,12,18, 9,15)( 8,13,11,17,10,16)(19,33,24,35,22,31)(20,34,23,36,21,32)(25,28)(26,27)(29,30)$
6F	$6^{6}$	$54$	$6$	$30$	$( 1,34, 4,36, 5,32)( 2,33, 3,35, 6,31)( 7,29,12,26, 9,28)( 8,30,11,25,10,27)(13,22,16,24,17,19)(14,21,15,23,18,20)$

Malle's constant $a(G)$: $1/12$

magma:ConjugacyClasses(G);

Character table

		1A	2A	2B	2C	3A	3B	3C	3D	3E	3F	3G	3H	3I	3J	3K	6A	6B	6C	6D	6E	6F
	Size	1	9	27	27	2	2	4	6	6	6	6	12	12	12	12	18	18	18	18	54	54
	2 P	1A	1A	1A	1A	3A	3B	3C	3D	3E	3F	3G	3H	3I	3J	3K	3D	3E	3A	3F	3G	3B
	3 P	1A	2A	2B	2C	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	2A	2A	2A	2A	2B	2C
	Type
324.121.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
324.121.1b	R	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$
324.121.1c	R	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$- 1$
324.121.1d	R	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$
324.121.2a	R	$2$	$0$	$2$	$0$	$2$	$2$	$2$	$2$	$2$	$2$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$0$	$0$	$0$	$0$	$- 1$	$0$
324.121.2b	R	$2$	$2$	$0$	$0$	$- 1$	$2$	$- 1$	$- 1$	$- 1$	$2$	$2$	$- 1$	$- 1$	$- 1$	$2$	$- 1$	$2$	$- 1$	$- 1$	$0$	$0$
324.121.2c	R	$2$	$2$	$0$	$0$	$- 1$	$2$	$- 1$	$- 1$	$2$	$- 1$	$2$	$- 1$	$- 1$	$2$	$- 1$	$- 1$	$- 1$	$2$	$- 1$	$0$	$0$
324.121.2d	R	$2$	$2$	$0$	$0$	$- 1$	$2$	$- 1$	$2$	$- 1$	$- 1$	$2$	$2$	$- 1$	$- 1$	$- 1$	$2$	$- 1$	$- 1$	$- 1$	$0$	$0$
324.121.2e	R	$2$	$2$	$0$	$0$	$2$	$2$	$2$	$- 1$	$- 1$	$- 1$	$2$	$- 1$	$2$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$2$	$0$	$0$
324.121.2f	R	$2$	$- 2$	$0$	$0$	$- 1$	$2$	$- 1$	$- 1$	$- 1$	$2$	$2$	$- 1$	$- 1$	$- 1$	$2$	$1$	$- 2$	$1$	$1$	$0$	$0$
324.121.2g	R	$2$	$- 2$	$0$	$0$	$- 1$	$2$	$- 1$	$- 1$	$2$	$- 1$	$2$	$- 1$	$- 1$	$2$	$- 1$	$1$	$1$	$- 2$	$1$	$0$	$0$
324.121.2h	R	$2$	$- 2$	$0$	$0$	$- 1$	$2$	$- 1$	$2$	$- 1$	$- 1$	$2$	$2$	$- 1$	$- 1$	$- 1$	$- 2$	$1$	$1$	$1$	$0$	$0$
324.121.2i	R	$2$	$- 2$	$0$	$0$	$2$	$2$	$2$	$- 1$	$- 1$	$- 1$	$2$	$- 1$	$2$	$- 1$	$- 1$	$1$	$1$	$1$	$- 2$	$0$	$0$
324.121.2j	R	$2$	$0$	$- 2$	$0$	$2$	$2$	$2$	$2$	$2$	$2$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$0$	$0$	$0$	$0$	$1$	$0$
324.121.4a	R	$4$	$0$	$0$	$0$	$- 2$	$4$	$- 2$	$- 2$	$- 2$	$4$	$- 2$	$1$	$1$	$1$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$
324.121.4b	R	$4$	$0$	$0$	$0$	$- 2$	$4$	$- 2$	$- 2$	$4$	$- 2$	$- 2$	$1$	$1$	$- 2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$
324.121.4c	R	$4$	$0$	$0$	$0$	$- 2$	$4$	$- 2$	$4$	$- 2$	$- 2$	$- 2$	$- 2$	$1$	$1$	$1$	$0$	$0$	$0$	$0$	$0$	$0$
324.121.4d	R	$4$	$0$	$0$	$0$	$4$	$4$	$4$	$- 2$	$- 2$	$- 2$	$- 2$	$1$	$- 2$	$1$	$1$	$0$	$0$	$0$	$0$	$0$	$0$
324.121.6a	R	$6$	$0$	$0$	$2$	$6$	$- 3$	$- 3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$
324.121.6b	R	$6$	$0$	$0$	$- 2$	$6$	$- 3$	$- 3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$1$
324.121.12a	R	$12$	$0$	$0$	$0$	$- 6$	$- 6$	$3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

magma:CharacterTable(G);

Regular extensions

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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	36T509

Degree	$36$
Order	$324$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_3^2:S_3^2$