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Group action invariants

Degree $n$:		$18$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:		$48$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:		$C_3^2:A_4$
Parity:		$1$	magma: IsEven(G);
Primitive:		no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:		$6$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:		(1,9,18)(2,10,17)(3,12,13)(4,11,14)(5,7,16)(6,8,15), (1,13,10)(2,14,9)(3,16,11)(4,15,12)(5,18,8)(6,17,7)	magma: Generators(G);

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$3$:	$C_3$ x 4
$9$:	$C_3^2$
$12$:	$A_4$
$27$:	$C_3^2:C_3$
$36$:	$C_3\times A_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$

Degree 6: $A_4$

Degree 9: $C_3^2:C_3$

Low degree siblings

18T48 x 2, 36T84, 36T97 x 3

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
	$1^{18}$	$1$	$1$	$()$
	$2^{6},1^{6}$	$3$	$2$	$( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)$
	$3^{4},1^{6}$	$3$	$3$	$( 7, 9,12)( 8,10,11)(13,18,16)(14,17,15)$
	$6^{2},1^{6}$	$3$	$6$	$( 7,10,12, 8, 9,11)(13,17,16,14,18,15)$
	$6^{2},1^{6}$	$3$	$6$	$( 7,11, 9, 8,12,10)(13,15,18,14,16,17)$
	$3^{4},1^{6}$	$3$	$3$	$( 7,12, 9)( 8,11,10)(13,16,18)(14,15,17)$
	$6,3^{2},2^{3}$	$3$	$6$	$( 1, 2)( 3, 4)( 5, 6)( 7, 9,12)( 8,10,11)(13,17,16,14,18,15)$
	$6,3^{2},2^{3}$	$3$	$6$	$( 1, 2)( 3, 4)( 5, 6)( 7,10,12, 8, 9,11)(13,18,16)(14,17,15)$
	$6,3^{2},2^{3}$	$3$	$6$	$( 1, 2)( 3, 4)( 5, 6)( 7,11, 9, 8,12,10)(13,16,18)(14,15,17)$
	$6,3^{2},2^{3}$	$3$	$6$	$( 1, 2)( 3, 4)( 5, 6)( 7,12, 9)( 8,11,10)(13,15,18,14,16,17)$
	$3^{6}$	$1$	$3$	$( 1, 3, 5)( 2, 4, 6)( 7, 9,12)( 8,10,11)(13,16,18)(14,15,17)$
	$6^{2},3^{2}$	$3$	$6$	$( 1, 3, 5)( 2, 4, 6)( 7,10,12, 8, 9,11)(13,15,18,14,16,17)$
	$6^{2},3^{2}$	$3$	$6$	$( 1, 5, 3)( 2, 6, 4)( 7,11, 9, 8,12,10)(13,17,16,14,18,15)$
	$3^{6}$	$1$	$3$	$( 1, 5, 3)( 2, 6, 4)( 7,12, 9)( 8,11,10)(13,18,16)(14,17,15)$
	$3^{6}$	$12$	$3$	$( 1, 7,13)( 2, 8,14)( 3, 9,16)( 4,10,15)( 5,12,18)( 6,11,17)$
	$3^{6}$	$12$	$3$	$( 1, 7,15)( 2, 8,16)( 3, 9,17)( 4,10,18)( 5,12,14)( 6,11,13)$
	$3^{6}$	$12$	$3$	$( 1, 7,17)( 2, 8,18)( 3, 9,14)( 4,10,13)( 5,12,15)( 6,11,16)$
	$3^{6}$	$12$	$3$	$( 1,13, 7)( 2,14, 8)( 3,16, 9)( 4,15,10)( 5,18,12)( 6,17,11)$
	$3^{6}$	$12$	$3$	$( 1,13,12)( 2,14,11)( 3,16, 7)( 4,15, 8)( 5,18, 9)( 6,17,10)$
	$3^{6}$	$12$	$3$	$( 1,13, 9)( 2,14,10)( 3,16,12)( 4,15,11)( 5,18, 7)( 6,17, 8)$

Group invariants

		1A	2A	3A1	3A-1	3B1	3B-1	3C1	3C-1	3D1	3D-1	3E1	3E-1	6A1	6A-1	6B1	6B-1	6C1	6C-1	6D1	6D-1
	Size	1	3	1	1	3	3	12	12	12	12	12	12	3	3	3	3	3	3	3	3
	2 P	1A	1A	3A-1	3A1	3B-1	3B1	3C-1	3C1	3D1	3E1	3E-1	3D-1	3B-1	3B-1	3B1	3A-1	3B-1	3A1	3B1	3B1
	3 P	1A	2A	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	2A	2A	2A	2A	2A	2A	2A	2A
	Type
108.22.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
108.22.1b1	C	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$
108.22.1b2	C	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$
108.22.1c1	C	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$
108.22.1c2	C	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$
108.22.1d1	C	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$
108.22.1d2	C	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$
108.22.1e1	C	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
108.22.1e2	C	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
108.22.3a	R	$3$	$-1$	$3$	$3$	$3$	$3$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$
108.22.3b1	C	$3$	$3$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$0$	$0$	$0$	$0$	$0$	$0$
108.22.3b2	C	$3$	$3$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
108.22.3c1	C	$3$	$-1$	$3$	$3$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$
108.22.3c2	C	$3$	$-1$	$3$	$3$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$0$	$0$	$0$	$0$	$0$	$0$	$-1$	$-1$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$
108.22.3d1	C	$3$	$-1$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$2{\zeta }_3$	$2{\zeta }_3^{-1}$	$2{\zeta }_3$	$2{\zeta }_3^{-1}$	$2$	$2$
108.22.3d2	C	$3$	$-1$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$2{\zeta }_3^{-1}$	$2{\zeta }_3$	$2{\zeta }_3^{-1}$	$2{\zeta }_3$	$2$	$2$
108.22.3e1	C	$3$	$-1$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$2{\zeta }_3^{-1}$	$2{\zeta }_3$	$2$	$2$	$2{\zeta }_3^{-1}$	$2{\zeta }_3$
108.22.3e2	C	$3$	$-1$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$2{\zeta }_3$	$2{\zeta }_3^{-1}$	$2$	$2$	$2{\zeta }_3$	$2{\zeta }_3^{-1}$
108.22.3f1	C	$3$	$-1$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$2$	$2$	$2{\zeta }_3^{-1}$	$2{\zeta }_3$	$2{\zeta }_3$	$2{\zeta }_3^{-1}$
108.22.3f2	C	$3$	$-1$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$2$	$2$	$2{\zeta }_3$	$2{\zeta }_3^{-1}$	$2{\zeta }_3^{-1}$	$2{\zeta }_3$

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