LMFDB - Galois group: 18T135

Show commands: Magma

magma:G := TransitiveGroup(18, 135);

Group invariants

Abstract group:	$C_3^2:S_3^2$	magma:IdentifyGroup(G);
Order:	$324=2^{2} \cdot 3^{4}$	magma:Order(G);
Cyclic:	no	magma:IsCyclic(G);
Abelian:	no	magma:IsAbelian(G);
Solvable:	yes	magma:IsSolvable(G);
Nilpotency class:	not nilpotent	magma:NilpotencyClass(G);

Group action invariants

Degree $n$:	$18$	magma:t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$135$	magma:t, n := TransitiveGroupIdentification(G); t;
Parity:	$-1$	magma:IsEven(G);
Primitive:	no	magma:IsPrimitive(G);
$\card{\Aut(F/K)}$:	$3$	magma:Order(Centralizer(SymmetricGroup(n), G));
Generators:	$(1,12,2,10,3,11)(4,18,6,16,5,17)(7,14,8,15,9,13)$, $(1,11,5,13,8,17)(2,12,4,14,9,18)(3,10,6,15,7,16)$, $(1,15,3,14,2,13)(4,11,5,10,6,12)(7,18,9,17,8,16)$	magma:Generators(G);

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$2$: $C_2$ x 3
$4$: $C_2^2$
$6$: $S_3$ x 5
$12$: $D_{6}$ x 5
$18$: $C_3^2:C_2$
$36$: $S_3^2$ x 4, 18T12
$54$: $(C_3^2:C_3):C_2$
$108$: 18T52, 18T58

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$6$:	$S_3$ x 5
$12$:	$D_{6}$ x 5
$18$:	$C_3^2:C_2$
$36$:	$S_3^2$ x 4, 18T12
$54$:	$(C_3^2:C_3):C_2$
$108$:	18T52, 18T58

Subfields

Degree 2: $C_2$
Degree 3: $S_3$
Degree 6: $D_{6}$
Degree 9: None

Low degree siblings

18T135 x 3, 27T118 x 4, 36T507 x 4
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	$2^{9}$	$3$	$2$	$9$	$( 1,15)( 2,13)( 3,14)( 4,17)( 5,16)( 6,18)( 7,12)( 8,10)( 9,11)$
2B	$2^{6},1^{6}$	$9$	$2$	$6$	$( 4, 8)( 5, 7)( 6, 9)(10,17)(11,18)(12,16)$
2C	$2^{9}$	$27$	$2$	$9$	$( 1,10)( 2,11)( 3,12)( 4,16)( 5,18)( 6,17)( 7,15)( 8,13)( 9,14)$
3A1	$3^{6}$	$1$	$3$	$12$	$( 1, 2, 3)( 4, 6, 5)( 7, 8, 9)(10,11,12)(13,14,15)(16,17,18)$
3A-1	$3^{6}$	$1$	$3$	$12$	$( 1, 3, 2)( 4, 5, 6)( 7, 9, 8)(10,12,11)(13,15,14)(16,18,17)$
3B	$3^{6}$	$2$	$3$	$12$	$( 1, 3, 2)( 4, 5, 6)( 7, 9, 8)(10,11,12)(13,14,15)(16,17,18)$
3C1	$3^{3},1^{9}$	$2$	$3$	$6$	$(10,12,11)(13,15,14)(16,18,17)$
3C-1	$3^{3},1^{9}$	$2$	$3$	$6$	$(10,11,12)(13,14,15)(16,17,18)$
3D	$3^{6}$	$6$	$3$	$12$	$( 1, 8, 5)( 2, 9, 4)( 3, 7, 6)(10,16,15)(11,17,13)(12,18,14)$
3E	$3^{6}$	$6$	$3$	$12$	$( 1, 8, 6)( 2, 9, 5)( 3, 7, 4)(10,18,15)(11,16,13)(12,17,14)$
3F	$3^{6}$	$6$	$3$	$12$	$( 1, 4, 8)( 2, 6, 9)( 3, 5, 7)(10,15,17)(11,13,18)(12,14,16)$
3G	$3^{4},1^{6}$	$6$	$3$	$8$	$( 1, 3, 2)( 7, 8, 9)(10,11,12)(13,15,14)$
3H	$3^{6}$	$12$	$3$	$12$	$( 1, 7, 4)( 2, 8, 6)( 3, 9, 5)(10,16,15)(11,17,13)(12,18,14)$
3I	$3^{6}$	$12$	$3$	$12$	$( 1, 4, 9)( 2, 6, 7)( 3, 5, 8)(10,13,17)(11,14,18)(12,15,16)$
3J	$3^{4},1^{6}$	$12$	$3$	$8$	$( 1, 2, 3)( 7, 9, 8)(13,15,14)(16,17,18)$
3K	$3^{6}$	$12$	$3$	$12$	$( 1, 7, 6)( 2, 8, 5)( 3, 9, 4)(10,18,14)(11,16,15)(12,17,13)$
6A1	$6^{3}$	$3$	$6$	$15$	$( 1,14, 2,15, 3,13)( 4,16, 6,17, 5,18)( 7,11, 8,12, 9,10)$
6A-1	$6^{3}$	$3$	$6$	$15$	$( 1,13, 3,15, 2,14)( 4,18, 5,17, 6,16)( 7,10, 9,12, 8,11)$
6B1	$6^{2},3^{2}$	$9$	$6$	$14$	$( 1, 2, 3)( 4, 9, 5, 8, 6, 7)(10,18,12,17,11,16)(13,14,15)$
6B-1	$6^{2},3^{2}$	$9$	$6$	$14$	$( 1, 3, 2)( 4, 7, 6, 8, 5, 9)(10,16,11,17,12,18)(13,15,14)$
6C	$6^{3}$	$18$	$6$	$15$	$( 1,18, 8,14, 5,12)( 2,16, 9,15, 4,10)( 3,17, 7,13, 6,11)$
6D	$6^{3}$	$18$	$6$	$15$	$( 1,16, 8,13, 6,11)( 2,17, 9,14, 5,12)( 3,18, 7,15, 4,10)$
6E	$6^{2},3^{2}$	$18$	$6$	$14$	$( 1, 5, 3, 6, 2, 4)( 7, 8, 9)(10,12,11)(13,18,14,16,15,17)$
6F	$6^{3}$	$18$	$6$	$15$	$( 1,10, 4,15, 8,17)( 2,11, 6,13, 9,18)( 3,12, 5,14, 7,16)$
6G	$6^{2},2^{3}$	$18$	$6$	$13$	$( 1,15, 3,14, 2,13)( 4,16)( 5,18)( 6,17)( 7,10, 8,11, 9,12)$
6H1	$6,3,2^{3},1^{3}$	$18$	$6$	$10$	$( 1, 7)( 2, 8)( 3, 9)(10,14,12,13,11,15)(16,17,18)$
6H-1	$6,3,2^{3},1^{3}$	$18$	$6$	$10$	$( 1, 3, 2)( 4, 8, 6, 9, 5, 7)(10,16)(11,17)(12,18)$
6I1	$6^{3}$	$27$	$6$	$15$	$( 1,12, 2,10, 3,11)( 4,18, 6,16, 5,17)( 7,14, 8,15, 9,13)$
6I-1	$6^{3}$	$27$	$6$	$15$	$( 1,11, 3,10, 2,12)( 4,17, 5,16, 6,18)( 7,13, 9,15, 8,14)$

Malle's constant $a(G)$: $1/6$

magma:ConjugacyClasses(G);

Character table

		1A	2A	2B	2C	3A1	3A-1	3B	3C1	3C-1	3D	3E	3F	3G	3H	3I	3J	3K	6A1	6A-1	6B1	6B-1	6C	6D	6E	6F	6G	6H1	6H-1	6I1	6I-1
	Size	1	3	9	27	1	1	2	2	2	6	6	6	6	12	12	12	12	3	3	9	9	18	18	18	18	18	18	18	27	27
	2 P	1A	1A	1A	1A	3A-1	3A1	3B	3C-1	3C1	3D	3E	3F	3G	3H	3I	3J	3K	3A1	3A-1	3A-1	3A1	3D	3E	3B	3F	3G	3C1	3C-1	3A1	3A-1
	3 P	1A	2A	2B	2C	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	2A	2A	2B	2B	2A	2A	2B	2A	2A	2B	2B	2C	2C
	Type
324.122.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
324.122.1b	R	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$
324.122.1c	R	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- 1$
324.122.1d	R	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$
324.122.2a	R	$2$	$0$	$2$	$0$	$2$	$2$	$- 1$	$- 1$	$- 1$	$2$	$2$	$2$	$2$	$- 1$	$- 1$	$- 1$	$- 1$	$0$	$0$	$2$	$2$	$- 1$	$0$	$0$	$0$	$- 1$	$- 1$	$0$	$0$	$0$
324.122.2b	R	$2$	$2$	$0$	$0$	$2$	$2$	$2$	$2$	$2$	$- 1$	$- 1$	$- 1$	$2$	$- 1$	$- 1$	$2$	$- 1$	$2$	$2$	$0$	$0$	$0$	$- 1$	$- 1$	$2$	$0$	$0$	$- 1$	$0$	$0$
324.122.2c	R	$2$	$2$	$0$	$0$	$2$	$2$	$2$	$2$	$2$	$- 1$	$- 1$	$2$	$- 1$	$- 1$	$- 1$	$- 1$	$2$	$2$	$2$	$0$	$0$	$0$	$- 1$	$2$	$- 1$	$0$	$0$	$- 1$	$0$	$0$
324.122.2d	R	$2$	$2$	$0$	$0$	$2$	$2$	$2$	$2$	$2$	$- 1$	$2$	$- 1$	$- 1$	$- 1$	$2$	$- 1$	$- 1$	$2$	$2$	$0$	$0$	$0$	$- 1$	$- 1$	$- 1$	$0$	$0$	$2$	$0$	$0$
324.122.2e	R	$2$	$2$	$0$	$0$	$2$	$2$	$2$	$2$	$2$	$2$	$- 1$	$- 1$	$- 1$	$2$	$- 1$	$- 1$	$- 1$	$2$	$2$	$0$	$0$	$0$	$2$	$- 1$	$- 1$	$0$	$0$	$- 1$	$0$	$0$
324.122.2f	R	$2$	$- 2$	$0$	$0$	$2$	$2$	$2$	$2$	$2$	$- 1$	$- 1$	$- 1$	$2$	$- 1$	$- 1$	$2$	$- 1$	$- 2$	$- 2$	$0$	$0$	$0$	$1$	$1$	$- 2$	$0$	$0$	$1$	$0$	$0$
324.122.2g	R	$2$	$- 2$	$0$	$0$	$2$	$2$	$2$	$2$	$2$	$- 1$	$- 1$	$2$	$- 1$	$- 1$	$- 1$	$- 1$	$2$	$- 2$	$- 2$	$0$	$0$	$0$	$1$	$- 2$	$1$	$0$	$0$	$1$	$0$	$0$
324.122.2h	R	$2$	$- 2$	$0$	$0$	$2$	$2$	$2$	$2$	$2$	$- 1$	$2$	$- 1$	$- 1$	$- 1$	$2$	$- 1$	$- 1$	$- 2$	$- 2$	$0$	$0$	$0$	$1$	$1$	$1$	$0$	$0$	$- 2$	$0$	$0$
324.122.2i	R	$2$	$- 2$	$0$	$0$	$2$	$2$	$2$	$2$	$2$	$2$	$- 1$	$- 1$	$- 1$	$2$	$- 1$	$- 1$	$- 1$	$- 2$	$- 2$	$0$	$0$	$0$	$- 2$	$1$	$1$	$0$	$0$	$1$	$0$	$0$
324.122.2j	R	$2$	$0$	$- 2$	$0$	$2$	$2$	$- 1$	$- 1$	$- 1$	$2$	$2$	$2$	$2$	$- 1$	$- 1$	$- 1$	$- 1$	$0$	$0$	$- 2$	$- 2$	$1$	$0$	$0$	$0$	$1$	$1$	$0$	$0$	$0$
324.122.3a1	C	$3$	$3$	$1$	$1$	$3 ζ_{3}^{- 1}$	$3 ζ_{3}$	$3$	$3 ζ_{3}^{- 1}$	$3 ζ_{3}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$3 ζ_{3}$	$3 ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$0$	$0$	$0$	$ζ_{3}^{- 1}$	$1$	$0$	$ζ_{3}$	$ζ_{3}^{- 1}$
324.122.3a2	C	$3$	$3$	$1$	$1$	$3 ζ_{3}$	$3 ζ_{3}^{- 1}$	$3$	$3 ζ_{3}$	$3 ζ_{3}^{- 1}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$3 ζ_{3}^{- 1}$	$3 ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$0$	$0$	$0$	$ζ_{3}$	$1$	$0$	$ζ_{3}^{- 1}$	$ζ_{3}$
324.122.3b1	C	$3$	$3$	$- 1$	$- 1$	$3 ζ_{3}^{- 1}$	$3 ζ_{3}$	$3$	$3 ζ_{3}^{- 1}$	$3 ζ_{3}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$3 ζ_{3}$	$3 ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}$	$0$	$0$	$0$	$- ζ_{3}^{- 1}$	$- 1$	$0$	$- ζ_{3}$	$- ζ_{3}^{- 1}$
324.122.3b2	C	$3$	$3$	$- 1$	$- 1$	$3 ζ_{3}$	$3 ζ_{3}^{- 1}$	$3$	$3 ζ_{3}$	$3 ζ_{3}^{- 1}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$3 ζ_{3}^{- 1}$	$3 ζ_{3}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$0$	$0$	$0$	$- ζ_{3}$	$- 1$	$0$	$- ζ_{3}^{- 1}$	$- ζ_{3}$
324.122.3c1	C	$3$	$- 3$	$- 1$	$1$	$3 ζ_{3}^{- 1}$	$3 ζ_{3}$	$3$	$3 ζ_{3}^{- 1}$	$3 ζ_{3}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 3 ζ_{3}$	$- 3 ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$- ζ_{3}$	$0$	$0$	$0$	$- ζ_{3}^{- 1}$	$- 1$	$0$	$ζ_{3}$	$ζ_{3}^{- 1}$
324.122.3c2	C	$3$	$- 3$	$- 1$	$1$	$3 ζ_{3}$	$3 ζ_{3}^{- 1}$	$3$	$3 ζ_{3}$	$3 ζ_{3}^{- 1}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 3 ζ_{3}^{- 1}$	$- 3 ζ_{3}$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$0$	$0$	$0$	$- ζ_{3}$	$- 1$	$0$	$ζ_{3}^{- 1}$	$ζ_{3}$
324.122.3d1	C	$3$	$- 3$	$1$	$- 1$	$3 ζ_{3}^{- 1}$	$3 ζ_{3}$	$3$	$3 ζ_{3}^{- 1}$	$3 ζ_{3}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 3 ζ_{3}$	$- 3 ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$ζ_{3}$	$ζ_{3}$	$0$	$0$	$0$	$ζ_{3}^{- 1}$	$1$	$0$	$- ζ_{3}$	$- ζ_{3}^{- 1}$
324.122.3d2	C	$3$	$- 3$	$1$	$- 1$	$3 ζ_{3}$	$3 ζ_{3}^{- 1}$	$3$	$3 ζ_{3}$	$3 ζ_{3}^{- 1}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 3 ζ_{3}^{- 1}$	$- 3 ζ_{3}$	$ζ_{3}$	$ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$0$	$0$	$0$	$ζ_{3}$	$1$	$0$	$- ζ_{3}^{- 1}$	$- ζ_{3}$
324.122.4a	R	$4$	$0$	$0$	$0$	$4$	$4$	$- 2$	$- 2$	$- 2$	$- 2$	$- 2$	$- 2$	$4$	$1$	$1$	$- 2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
324.122.4b	R	$4$	$0$	$0$	$0$	$4$	$4$	$- 2$	$- 2$	$- 2$	$- 2$	$- 2$	$4$	$- 2$	$1$	$1$	$1$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
324.122.4c	R	$4$	$0$	$0$	$0$	$4$	$4$	$- 2$	$- 2$	$- 2$	$- 2$	$4$	$- 2$	$- 2$	$1$	$- 2$	$1$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
324.122.4d	R	$4$	$0$	$0$	$0$	$4$	$4$	$- 2$	$- 2$	$- 2$	$4$	$- 2$	$- 2$	$- 2$	$- 2$	$1$	$1$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
324.122.6a1	C	$6$	$0$	$2$	$0$	$6 ζ_{3}^{- 1}$	$6 ζ_{3}$	$- 3$	$- 3 ζ_{3}^{- 1}$	$- 3 ζ_{3}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$2 ζ_{3}^{- 1}$	$2 ζ_{3}$	$- ζ_{3}$	$0$	$0$	$0$	$- ζ_{3}^{- 1}$	$- 1$	$0$	$0$	$0$
324.122.6a2	C	$6$	$0$	$2$	$0$	$6 ζ_{3}$	$6 ζ_{3}^{- 1}$	$- 3$	$- 3 ζ_{3}$	$- 3 ζ_{3}^{- 1}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$2 ζ_{3}$	$2 ζ_{3}^{- 1}$	$- ζ_{3}^{- 1}$	$0$	$0$	$0$	$- ζ_{3}$	$- 1$	$0$	$0$	$0$
324.122.6b1	C	$6$	$0$	$- 2$	$0$	$6 ζ_{3}^{- 1}$	$6 ζ_{3}$	$- 3$	$- 3 ζ_{3}^{- 1}$	$- 3 ζ_{3}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 2 ζ_{3}^{- 1}$	$- 2 ζ_{3}$	$ζ_{3}$	$0$	$0$	$0$	$ζ_{3}^{- 1}$	$1$	$0$	$0$	$0$
324.122.6b2	C	$6$	$0$	$- 2$	$0$	$6 ζ_{3}$	$6 ζ_{3}^{- 1}$	$- 3$	$- 3 ζ_{3}$	$- 3 ζ_{3}^{- 1}$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 2 ζ_{3}$	$- 2 ζ_{3}^{- 1}$	$ζ_{3}^{- 1}$	$0$	$0$	$0$	$ζ_{3}$	$1$	$0$	$0$	$0$

magma:CharacterTable(G);

Regular extensions

Data not computed

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	18T135

Degree	$18$
Order	$324$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_3^2:S_3^2$