Galois group: 18T49

• Label: 18T49
• Degree: $18$
• Order: $108$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $\He_3:C_4$

Group invariants

Abstract group:		$\He_3:C_4$
Order:		$108=2^{2} \cdot 3^{3}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$18$
Transitive number $t$:		$49$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$3$
Generators:		$(1,8,14)(2,9,15)(3,7,13)(4,10,18)(5,11,16)(6,12,17)$, $(1,6,15,10,3,5,14,12,2,4,13,11)(7,17,8,18,9,16)$

Low degree resolvents

$\card{(G/N)}$	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 2: $C_2$

Degree 3: None

Degree 6: $C_3^2:C_4$

Degree 9: None

Low degree siblings

18T49, 27T32, 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	Index	Representative
1A	$1^{18}$	$1$	$1$	$0$	$()$
2A	$2^{6},1^{6}$	$9$	$2$	$6$	$( 1, 8)( 2, 9)( 3, 7)(10,18)(11,16)(12,17)$
3A1	$3^{6}$	$1$	$3$	$12$	$( 1, 3, 2)( 4, 6, 5)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)$
3A-1	$3^{6}$	$1$	$3$	$12$	$( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)$
3B	$3^{5},1^{3}$	$12$	$3$	$10$	$( 1, 9,14)( 2, 7,15)( 3, 8,13)( 4, 5, 6)(10,12,11)$
3C	$3^{6}$	$12$	$3$	$12$	$( 1, 7,14)( 2, 8,15)( 3, 9,13)( 4,16,10)( 5,17,11)( 6,18,12)$
4A1	$4^{3},2^{3}$	$9$	$4$	$12$	$( 1,10, 8,18)( 2,11, 9,16)( 3,12, 7,17)( 4,13)( 5,14)( 6,15)$
4A-1	$4^{3},2^{3}$	$9$	$4$	$12$	$( 1,18, 8,10)( 2,16, 9,11)( 3,17, 7,12)( 4,13)( 5,14)( 6,15)$
6A1	$6^{2},3^{2}$	$9$	$6$	$14$	$( 1, 9, 3, 8, 2, 7)( 4, 5, 6)(10,16,12,18,11,17)(13,14,15)$
6A-1	$6^{2},3^{2}$	$9$	$6$	$14$	$( 1, 7, 2, 8, 3, 9)( 4, 6, 5)(10,17,11,18,12,16)(13,15,14)$
12A1	$12,6$	$9$	$12$	$16$	$( 1,17, 9,10, 3,16, 8,12, 2,18, 7,11)( 4,15, 5,13, 6,14)$
12A-1	$12,6$	$9$	$12$	$16$	$( 1, 6,15,18, 2, 4,13,16, 3, 5,14,17)( 7,12, 9,11, 8,10)$
12A5	$12,6$	$9$	$12$	$16$	$( 1,16, 7,10, 2,17, 8,11, 3,18, 9,12)( 4,14, 6,13, 5,15)$
12A-5	$12,6$	$9$	$12$	$16$	$( 1, 4,14,18, 3, 6,13,17, 2, 5,15,16)( 7,10, 8,11, 9,12)$

Malle's constant $a(G)$:     $1/6$

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	$4$	$0$	$4$	$4$	$1$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed