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Group action invariants

Degree $n$:		$27$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:		$21$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:		$C_3\wr C_3$
Parity:		$1$	magma: IsEven(G);
Primitive:		no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:		$3$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:		(4,5,6)(7,15,10)(8,13,11)(9,14,12)(16,20,22)(17,21,23)(18,19,24)(25,27,26), (1,11,19)(2,12,20)(3,10,21)(4,15,22)(5,13,23)(6,14,24)(7,16,25)(8,17,26)(9,18,27)	magma: Generators(G);

Low degree resolvents

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

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: $C_3$

Degree 9: $C_3^2:C_3$

Low degree siblings

9T17 x 3, 27T19, 27T27 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
	$1^{27}$	$1$	$1$	$()$
	$3^{8},1^{3}$	$9$	$3$	$( 4, 5, 6)( 7,15,10)( 8,13,11)( 9,14,12)(16,20,22)(17,21,23)(18,19,24) (25,27,26)$
	$3^{8},1^{3}$	$9$	$3$	$( 4, 6, 5)( 7,10,15)( 8,11,13)( 9,12,14)(16,22,20)(17,23,21)(18,24,19) (25,26,27)$
	$3^{9}$	$1$	$3$	$( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$
	$3^{9}$	$1$	$3$	$( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)$
	$3^{9}$	$3$	$3$	$( 1, 4,25)( 2, 5,26)( 3, 6,27)( 7,11,15)( 8,12,13)( 9,10,14)(16,19,22) (17,20,23)(18,21,24)$
	$9^{3}$	$9$	$9$	$( 1, 7,18, 3, 9,17, 2, 8,16)( 4,10,19, 6,12,21, 5,11,20)(13,23,27,15,22,26,14, 24,25)$
	$3^{9}$	$3$	$3$	$( 1, 7,22)( 2, 8,23)( 3, 9,24)( 4,11,16)( 5,12,17)( 6,10,18)(13,20,26) (14,21,27)(15,19,25)$
	$9^{3}$	$9$	$9$	$( 1, 7,19, 3, 9,21, 2, 8,20)( 4,12,24, 6,11,23, 5,10,22)(13,16,26,15,18,25,14, 17,27)$
	$3^{9}$	$3$	$3$	$( 1, 8,24)( 2, 9,22)( 3, 7,23)( 4,12,18)( 5,10,16)( 6,11,17)(13,21,25) (14,19,26)(15,20,27)$
	$3^{9}$	$3$	$3$	$( 1, 9,23)( 2, 7,24)( 3, 8,22)( 4,10,17)( 5,11,18)( 6,12,16)(13,19,27) (14,20,25)(15,21,26)$
	$3^{9}$	$3$	$3$	$( 1,16,15)( 2,17,13)( 3,18,14)( 4,19, 7)( 5,20, 8)( 6,21, 9)(10,27,24) (11,25,22)(12,26,23)$
	$9^{3}$	$9$	$9$	$( 1,16, 8, 2,17, 9, 3,18, 7)( 4,20,11, 5,21,12, 6,19,10)(13,25,24,14,26,22,15, 27,23)$
	$9^{3}$	$9$	$9$	$( 1,16,11, 2,17,12, 3,18,10)( 4,21,13, 5,19,14, 6,20,15)( 7,27,22, 8,25,23, 9, 26,24)$
	$3^{9}$	$3$	$3$	$( 1,17,14)( 2,18,15)( 3,16,13)( 4,20, 9)( 5,21, 7)( 6,19, 8)(10,25,23) (11,26,24)(12,27,22)$
	$3^{9}$	$3$	$3$	$( 1,18,13)( 2,16,14)( 3,17,15)( 4,21, 8)( 5,19, 9)( 6,20, 7)(10,26,22) (11,27,23)(12,25,24)$
	$3^{9}$	$3$	$3$	$( 1,25, 4)( 2,26, 5)( 3,27, 6)( 7,15,11)( 8,13,12)( 9,14,10)(16,22,19) (17,23,20)(18,24,21)$

Group invariants

		1A	3A1	3A-1	3B1	3B-1	3C1	3C-1	3D1	3D-1	3E1	3E-1	3F1	3F-1	9A1	9A-1	9B1	9B-1
	Size	1	1	1	3	3	3	3	3	3	3	3	9	9	9	9	9	9
	3 P	1A	3A-1	3A1	3B1	3E1	3C-1	3C1	3D1	3E-1	3D-1	3B-1	3F-1	3F1	9A-1	9A1	9B-1	9B1
	Type
81.7.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
81.7.1b1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$
81.7.1b2	C	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$
81.7.1c1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$
81.7.1c2	C	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$
81.7.1d1	C	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
81.7.1d2	C	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
81.7.1e1	C	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$
81.7.1e2	C	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3$
81.7.3a1	C	$3$	$3$	$3$	$0$	$0$	$0$	$0$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3a2	C	$3$	$3$	$3$	$0$	$0$	$0$	$0$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3b1	C	$3$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$-1+{\zeta }_3$	$-2-{\zeta }_3$	$1+2{\zeta }_3$	$-1-2{\zeta }_3$	$0$	$0$	$1-{\zeta }_3$	$2+{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3b2	C	$3$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$-2-{\zeta }_3$	$-1+{\zeta }_3$	$-1-2{\zeta }_3$	$1+2{\zeta }_3$	$0$	$0$	$2+{\zeta }_3$	$1-{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3c1	C	$3$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$-1-2{\zeta }_3$	$1+2{\zeta }_3$	$1-{\zeta }_3$	$2+{\zeta }_3$	$0$	$0$	$-2-{\zeta }_3$	$-1+{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3c2	C	$3$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$1+2{\zeta }_3$	$-1-2{\zeta }_3$	$2+{\zeta }_3$	$1-{\zeta }_3$	$0$	$0$	$-1+{\zeta }_3$	$-2-{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3d1	C	$3$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$2+{\zeta }_3$	$1-{\zeta }_3$	$-2-{\zeta }_3$	$-1+{\zeta }_3$	$0$	$0$	$1+2{\zeta }_3$	$-1-2{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$
81.7.3d2	C	$3$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$1-{\zeta }_3$	$2+{\zeta }_3$	$-1+{\zeta }_3$	$-2-{\zeta }_3$	$0$	$0$	$-1-2{\zeta }_3$	$1+2{\zeta }_3$	$0$	$0$	$0$	$0$	$0$	$0$

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