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Group action invariants

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

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$6$:	$S_3$ x 2
$12$:	$D_{6}$ x 2

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 3: $S_3$ x 2

Degree 4: $C_2^2$

Degree 6: $S_3$ x 2, $D_{6}$ x 4, $S_3^2$

Degree 9: $S_3^2$

Degree 12: $D_6$ x 2, $S_3^2$

Degree 18: $S_3^2$, $S_3^2$ x 2

Low degree siblings

6T9, 9T8, 12T16, 18T9, 18T11 x 2

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	Representative
	$1^{36}$	$1$	$1$	$()$
	$2^{18}$	$9$	$2$	$( 1, 2)( 3, 4)( 5,35)( 6,36)( 7,34)( 8,33)( 9,31)(10,32)(11,29)(12,30)(13,28) (14,27)(15,26)(16,25)(17,24)(18,23)(19,21)(20,22)$
	$2^{18}$	$3$	$2$	$( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,22)(10,21)(11,23)(12,24)(13,25)(14,26)(15,27) (16,28)(17,30)(18,29)(19,32)(20,31)(33,36)(34,35)$
	$2^{18}$	$3$	$2$	$( 1, 4)( 2, 3)( 5,34)( 6,33)( 7,35)( 8,36)( 9,20)(10,19)(11,18)(12,17)(13,16) (14,15)(21,32)(22,31)(23,29)(24,30)(25,28)(26,27)$
	$6^{6}$	$6$	$6$	$( 1, 5,33, 3, 7,36)( 2, 6,34, 4, 8,35)( 9,28,18,22,16,29)(10,27,17,21,15,30) (11,25,20,23,13,31)(12,26,19,24,14,32)$
	$3^{12}$	$2$	$3$	$( 1, 7,33)( 2, 8,34)( 3, 5,36)( 4, 6,35)( 9,16,18)(10,15,17)(11,13,20) (12,14,19)(21,27,30)(22,28,29)(23,25,31)(24,26,32)$
	$6^{6}$	$6$	$6$	$( 1,10,28,35,14,23)( 2, 9,27,36,13,24)( 3,11,26,34,16,21)( 4,12,25,33,15,22) ( 5,20,32, 8,18,30)( 6,19,31, 7,17,29)$
	$3^{12}$	$4$	$3$	$( 1,12,29)( 2,11,30)( 3, 9,32)( 4,10,31)( 5,16,24)( 6,15,23)( 7,14,22) ( 8,13,21)(17,25,35)(18,26,36)(19,28,33)(20,27,34)$
	$3^{12}$	$2$	$3$	$( 1,14,28)( 2,13,27)( 3,16,26)( 4,15,25)( 5,18,32)( 6,17,31)( 7,19,29) ( 8,20,30)( 9,24,36)(10,23,35)(11,21,34)(12,22,33)$

Group invariants

		1A	2A	2B	2C	3A	3B	3C	6A	6B
	Size	1	3	3	9	2	2	4	6	6
	2 P	1A	1A	1A	1A	3A	3B	3C	3A	3B
	3 P	1A	2A	2B	2C	1A	1A	1A	2A	2B
	Type
36.10.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
36.10.1b	R	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$
36.10.1c	R	$1$	$-1$	$1$	$-1$	$1$	$1$	$1$	$-1$	$1$
36.10.1d	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$-1$
36.10.2a	R	$2$	$0$	$2$	$0$	$2$	$-1$	$-1$	$0$	$-1$
36.10.2b	R	$2$	$2$	$0$	$0$	$-1$	$2$	$-1$	$-1$	$0$
36.10.2c	R	$2$	$-2$	$0$	$0$	$-1$	$2$	$-1$	$1$	$0$
36.10.2d	R	$2$	$0$	$-2$	$0$	$2$	$-1$	$-1$	$0$	$1$
36.10.4a	R	$4$	$0$	$0$	$0$	$-2$	$-2$	$1$	$0$	$0$

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