Galois group: 27T32

• Label: 27T32
• Degree: $27$
• Order: $108$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $\He_3:C_4$

Group action invariants

Degree $n$:		$27$
Transitive number $t$:		$32$
Group:		$\He_3:C_4$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$3$
Generators:		(1,15,9,4)(2,13,7,5)(3,14,8,6)(10,21,27,18)(11,19,25,16)(12,20,26,17), (1,25,22,18,3,27,24,17,2,26,23,16)(4,10,19,14,6,12,21,13,5,11,20,15)(7,9,8)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$
$36$:	$C_3^2:C_4$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 3: None

Degree 9: $C_3^2:C_4$

Low degree siblings

18T49 x 2, 36T85 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 4, 4, 4, 4, 4, 4, 1, 1, 1 $	$9$	$4$	$( 4,13,26,16)( 5,14,27,17)( 6,15,25,18)( 7,10,22,19)( 8,11,23,20)( 9,12,24,21)$
	$ 4, 4, 4, 4, 4, 4, 1, 1, 1 $	$9$	$4$	$( 4,16,26,13)( 5,17,27,14)( 6,18,25,15)( 7,19,22,10)( 8,20,23,11)( 9,21,24,12)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1 $	$9$	$2$	$( 4,26)( 5,27)( 6,25)( 7,22)( 8,23)( 9,24)(10,19)(11,20)(12,21)(13,16)(14,17) (15,18)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$1$	$3$	$( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)(19,20,21) (22,23,24)(25,26,27)$
	$ 12, 12, 3 $	$9$	$12$	$( 1, 2, 3)( 4,14,25,16, 5,15,26,17, 6,13,27,18)( 7,11,24,19, 8,12,22,20, 9,10, 23,21)$
	$ 12, 12, 3 $	$9$	$12$	$( 1, 2, 3)( 4,17,25,13, 5,18,26,14, 6,16,27,15)( 7,20,24,10, 8,21,22,11, 9,19, 23,12)$
	$ 6, 6, 6, 6, 3 $	$9$	$6$	$( 1, 2, 3)( 4,27, 6,26, 5,25)( 7,23, 9,22, 8,24)(10,20,12,19,11,21) (13,17,15,16,14,18)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$1$	$3$	$( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)(19,21,20) (22,24,23)(25,27,26)$
	$ 12, 12, 3 $	$9$	$12$	$( 1, 3, 2)( 4,15,27,16, 6,14,26,18, 5,13,25,17)( 7,12,23,19, 9,11,22,21, 8,10, 24,20)$
	$ 12, 12, 3 $	$9$	$12$	$( 1, 3, 2)( 4,18,27,13, 6,17,26,15, 5,16,25,14)( 7,21,23,10, 9,20,22,12, 8,19, 24,11)$
	$ 6, 6, 6, 6, 3 $	$9$	$6$	$( 1, 3, 2)( 4,25, 5,26, 6,27)( 7,24, 8,22, 9,23)(10,21,11,19,12,20) (13,18,14,16,15,17)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$12$	$3$	$( 1, 4,25)( 2, 5,26)( 3, 6,27)( 7,11,14)( 8,12,15)( 9,10,13)(16,20,23) (17,21,24)(18,19,22)$
	$ 3, 3, 3, 3, 3, 3, 3, 3, 3 $	$12$	$3$	$( 1, 7,23)( 2, 8,24)( 3, 9,22)( 4,10,17)( 5,11,18)( 6,12,16)(13,20,26) (14,21,27)(15,19,25)$

Group invariants

Order:		$108=2^{2} \cdot 3^{3}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent
Label:		108.15
Character table:

		1A	2A	3A1	3A-1	3B	3C	4A1	4A-1	6A1	6A-1	12A1	12A-1	12A5	12A-5
	Size	1	9	1	1	12	12	9	9	9	9	9	9	9	9
	2 P	1A	1A	3A-1	3A1	3B	3C	2A	2A	3A1	3A-1	6A1	6A-1	6A-1	6A1
	3 P	1A	2A	1A	1A	1A	1A	4A-1	4A1	2A	2A	4A1	4A-1	4A1	4A-1
	Type
108.15.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
108.15.1b	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
108.15.1c1	C	$1$	$-1$	$1$	$1$	$1$	$1$	$-i$	$i$	$-1$	$-1$	$i$	$-i$	$i$	$-i$
108.15.1c2	C	$1$	$-1$	$1$	$1$	$1$	$1$	$i$	$-i$	$-1$	$-1$	$-i$	$i$	$-i$	$i$
108.15.3a1	C	$3$	$-1$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$0$	$0$	$1$	$1$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
108.15.3a2	C	$3$	$-1$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$0$	$0$	$1$	$1$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
108.15.3b1	C	$3$	$-1$	$3{\zeta }_3^{-1}$	$3{\zeta }_3$	$0$	$0$	$-1$	$-1$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$
108.15.3b2	C	$3$	$-1$	$3{\zeta }_3$	$3{\zeta }_3^{-1}$	$0$	$0$	$-1$	$-1$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$
108.15.3c1	C	$3$	$1$	$-3{\zeta }_{12}^2$	$3{\zeta }_{12}^4$	$0$	$0$	$-{\zeta }_{12}^3$	${\zeta }_{12}^3$	${\zeta }_{12}^4$	$-{\zeta }_{12}^2$	$-{\zeta }_{12}^5$	${\zeta }_{12}$	$-{\zeta }_{12}$	${\zeta }_{12}^5$
108.15.3c2	C	$3$	$1$	$3{\zeta }_{12}^4$	$-3{\zeta }_{12}^2$	$0$	$0$	${\zeta }_{12}^3$	$-{\zeta }_{12}^3$	$-{\zeta }_{12}^2$	${\zeta }_{12}^4$	${\zeta }_{12}$	$-{\zeta }_{12}^5$	${\zeta }_{12}^5$	$-{\zeta }_{12}$
108.15.3c3	C	$3$	$1$	$-3{\zeta }_{12}^2$	$3{\zeta }_{12}^4$	$0$	$0$	${\zeta }_{12}^3$	$-{\zeta }_{12}^3$	${\zeta }_{12}^4$	$-{\zeta }_{12}^2$	${\zeta }_{12}^5$	$-{\zeta }_{12}$	${\zeta }_{12}$	$-{\zeta }_{12}^5$
108.15.3c4	C	$3$	$1$	$3{\zeta }_{12}^4$	$-3{\zeta }_{12}^2$	$0$	$0$	$-{\zeta }_{12}^3$	${\zeta }_{12}^3$	$-{\zeta }_{12}^2$	${\zeta }_{12}^4$	$-{\zeta }_{12}$	${\zeta }_{12}^5$	$-{\zeta }_{12}^5$	${\zeta }_{12}$
108.15.4a	R	$4$	$0$	$4$	$4$	$-2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
108.15.4b	R	$4$	$0$	$4$	$4$	$1$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$