LMFDB - Galois group: 36T1160

Show commands: Magma

magma:G := TransitiveGroup(36, 1160);

Group invariants

Abstract group:	$C_3^3:D_{12}$	magma:IdentifyGroup(G);
Order:	$648=2^{3} \cdot 3^{4}$	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 $n$:	$36$	magma:t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$1160$	magma:t, n := TransitiveGroupIdentification(G); t;
Parity:	$-1$	magma:IsEven(G);
Primitive:	no	magma:IsPrimitive(G);
$\card{\Aut(F/K)}$:	$1$	magma:Order(Centralizer(SymmetricGroup(n), G));
Generators:	$(1,17,13,28,25,4)(2,16,15,30,26,6)(3,18,14,29,27,5)(7,22,21,36,33,11)(8,23,20,35,31,12)(9,24,19,34,32,10)$, $(1,28,20,12,2,29,21,10,3,30,19,11)(4,7,24,26,5,9,22,27,6,8,23,25)(13,17,32,36,15,18,31,35,14,16,33,34)$	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 2
$8$: $D_{4}$
$12$: $D_{6}$ x 2
$24$: $D_{12}$, $(C_6\times C_2):C_2$
$36$: $S_3^2$
$72$: $C_3^2:D_4$, 12T38
$216$: 12T116, 12T118

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 2
$8$:	$D_{4}$
$12$:	$D_{6}$ x 2
$24$:	$D_{12}$, $(C_6\times C_2):C_2$
$36$:	$S_3^2$
$72$:	$C_3^2:D_4$, 12T38
$216$:	12T116, 12T118

Subfields

Degree 2: $C_2$
Degree 3: $S_3$
Degree 4: $D_{4}$
Degree 6: $D_{6}$
Degree 9: None
Degree 12: $(C_6\times C_2):C_2$, 12T118, 12T169 x 2
Degree 18: None

Low degree siblings

12T169 x 2, 18T210 x 2, 24T1521 x 2, 24T1533, 36T1105 x 2, 36T1106 x 2, 36T1107 x 2, 36T1153, 36T1154, 36T1167, 36T1173, 36T1190 x 2, 36T1195 x 2, 36T1225 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	$1^{36}$	$1$	$1$	$0$	$()$
2A	$2^{18}$	$9$	$2$	$18$	$( 1,21)( 2,19)( 3,20)( 4,24)( 5,22)( 6,23)( 7,27)( 8,26)( 9,25)(10,29)(11,30)(12,28)(13,31)(14,32)(15,33)(16,34)(17,36)(18,35)$
2B	$2^{18}$	$18$	$2$	$18$	$( 1,28)( 2,30)( 3,29)( 4,13)( 5,14)( 6,15)( 7,35)( 8,34)( 9,36)(10,20)(11,19)(12,21)(16,26)(17,25)(18,27)(22,32)(23,33)(24,31)$
2C	$2^{17},1^{2}$	$54$	$2$	$17$	$( 1,19)( 2,21)( 3,20)( 4,18)( 5,17)( 6,16)( 7,14)( 8,13)( 9,15)(11,12)(22,35)(23,36)(24,34)(25,33)(26,31)(27,32)(28,29)$
3A	$3^{12}$	$2$	$3$	$24$	$( 1,14,26)( 2,13,27)( 3,15,25)( 4,18,30)( 5,16,28)( 6,17,29)( 7,19,31)( 8,21,32)( 9,20,33)(10,23,36)(11,24,35)(12,22,34)$
3B	$3^{12}$	$2$	$3$	$24$	$( 1, 3, 2)( 4, 6, 5)( 7, 8, 9)(10,12,11)(13,14,15)(16,18,17)(19,21,20)(22,24,23)(25,27,26)(28,30,29)(31,32,33)(34,35,36)$
3C	$3^{6},1^{18}$	$4$	$3$	$12$	$( 1, 3, 2)( 7, 9, 8)(13,14,15)(19,20,21)(25,27,26)(31,33,32)$
3D	$3^{12}$	$4$	$3$	$24$	$( 1, 2, 3)( 4, 5, 6)( 7, 8, 9)(10,12,11)(13,15,14)(16,17,18)(19,21,20)(22,24,23)(25,26,27)(28,29,30)(31,32,33)(34,35,36)$
3E	$3^{12}$	$4$	$3$	$24$	$( 1,25,13)( 2,26,15)( 3,27,14)( 4,29,16)( 5,30,17)( 6,28,18)( 7,32,20)( 8,33,19)( 9,31,21)(10,34,24)(11,36,22)(12,35,23)$
3F1	$3^{12}$	$4$	$3$	$24$	$( 1,25,13)( 2,26,15)( 3,27,14)( 4,29,16)( 5,30,17)( 6,28,18)( 7,33,21)( 8,31,20)( 9,32,19)(10,35,22)(11,34,23)(12,36,24)$
3F-1	$3^{12}$	$4$	$3$	$24$	$( 1,13,25)( 2,15,26)( 3,14,27)( 4,16,29)( 5,17,30)( 6,18,28)( 7,21,33)( 8,20,31)( 9,19,32)(10,22,35)(11,23,34)(12,24,36)$
3G	$3^{12}$	$8$	$3$	$24$	$( 1,25,13)( 2,26,15)( 3,27,14)( 4,28,17)( 5,29,18)( 6,30,16)( 7,32,20)( 8,33,19)( 9,31,21)(10,36,23)(11,35,24)(12,34,22)$
3H	$3^{6},1^{18}$	$8$	$3$	$12$	$( 7, 8, 9)(10,12,11)(19,21,20)(22,24,23)(31,32,33)(34,35,36)$
3I	$3^{12}$	$8$	$3$	$24$	$( 1,25,13)( 2,26,15)( 3,27,14)( 4,30,18)( 5,28,16)( 6,29,17)( 7,33,21)( 8,31,20)( 9,32,19)(10,36,23)(11,35,24)(12,34,22)$
3J	$3^{9},1^{9}$	$8$	$3$	$18$	$( 4, 5, 6)( 7, 8, 9)(10,11,12)(16,17,18)(19,21,20)(22,23,24)(28,29,30)(31,32,33)(34,36,35)$
3K	$3^{12}$	$8$	$3$	$24$	$( 1,27,15)( 2,25,14)( 3,26,13)( 4,30,18)( 5,28,16)( 6,29,17)( 7,33,21)( 8,31,20)( 9,32,19)(10,34,24)(11,36,22)(12,35,23)$
3L1	$3^{12}$	$8$	$3$	$24$	$( 1,13,25)( 2,15,26)( 3,14,27)( 4,16,29)( 5,17,30)( 6,18,28)( 7,19,31)( 8,21,32)( 9,20,33)(10,23,36)(11,24,35)(12,22,34)$
3L-1	$3^{12}$	$8$	$3$	$24$	$( 1,13,25)( 2,15,26)( 3,14,27)( 4,18,30)( 5,16,28)( 6,17,29)( 7,19,31)( 8,21,32)( 9,20,33)(10,24,34)(11,22,36)(12,23,35)$
4A	$4^{9}$	$54$	$4$	$27$	$( 1,29,19,11)( 2,30,20,12)( 3,28,21,10)( 4, 9,22,27)( 5, 8,23,25)( 6, 7,24,26)(13,18,33,34)(14,17,31,35)(15,16,32,36)$
6A	$6^{6}$	$18$	$6$	$30$	$( 1, 8,14,21,26,32)( 2, 7,13,19,27,31)( 3, 9,15,20,25,33)( 4,11,18,24,30,35)( 5,12,16,22,28,34)( 6,10,17,23,29,36)$
6B	$6^{6}$	$18$	$6$	$30$	$( 1,20, 3,19, 2,21)( 4,23, 6,22, 5,24)( 7,27, 8,26, 9,25)(10,29,12,28,11,30)(13,32,14,33,15,31)(16,35,18,36,17,34)$
6C1	$6^{6}$	$18$	$6$	$30$	$( 1,16,14,28,26, 5)( 2,18,13,30,27, 4)( 3,17,15,29,25, 6)( 7,24,19,35,31,11)( 8,22,21,34,32,12)( 9,23,20,36,33,10)$
6C-1	$6^{6}$	$18$	$6$	$30$	$( 1, 6,26,29,14,17)( 2, 5,27,28,13,16)( 3, 4,25,30,15,18)( 7,12,31,34,19,22)( 8,10,32,36,21,23)( 9,11,33,35,20,24)$
6D	$6^{6}$	$36$	$6$	$30$	$( 1,11, 2,10, 3,12)( 4,31, 5,32, 6,33)( 7,16, 8,17, 9,18)(13,23,15,22,14,24)(19,28,21,29,20,30)(25,34,26,35,27,36)$
6E	$6^{6}$	$36$	$6$	$30$	$( 1,31,25,21,13, 9)( 2,33,26,19,15, 8)( 3,32,27,20,14, 7)( 4,35,29,23,16,12)( 5,34,30,24,17,10)( 6,36,28,22,18,11)$
6F1	$6^{6}$	$36$	$6$	$30$	$( 1,23,25,11,13,34)( 2,22,26,10,15,35)( 3,24,27,12,14,36)( 4, 7,29,33,16,21)( 5, 8,30,31,17,20)( 6, 9,28,32,18,19)$
6F-1	$6^{6}$	$36$	$6$	$30$	$( 1,34,13,11,25,23)( 2,35,15,10,26,22)( 3,36,14,12,27,24)( 4,21,16,33,29, 7)( 5,20,17,31,30, 8)( 6,19,18,32,28, 9)$
6G	$6^{3},2^{8},1^{2}$	$108$	$6$	$23$	$( 1,21, 3,19, 2,20)( 4,18)( 5,17)( 6,16)( 7,13, 9,14, 8,15)(11,12)(22,35)(23,36)(24,34)(25,31,27,33,26,32)(28,29)$
12A1	$12^{3}$	$54$	$12$	$33$	$( 1,10,20,29, 3,12,19,28, 2,11,21,30)( 4,26,23, 9, 6,25,22, 7, 5,27,24, 8)(13,35,32,18,14,36,33,17,15,34,31,16)$
12A5	$12^{3}$	$54$	$12$	$33$	$( 1,12,21,29, 2,10,19,30, 3,11,20,28)( 4,25,24, 9, 5,26,22, 8, 6,27,23, 7)(13,36,31,18,15,35,33,16,14,34,32,17)$

Malle's constant $a(G)$: $1/12$

magma:ConjugacyClasses(G);

Character table

		1A	2A	2B	2C	3A	3B	3C	3D	3E	3F1	3F-1	3G	3H	3I	3J	3K	3L1	3L-1	4A	6A	6B	6C1	6C-1	6D	6E	6F1	6F-1	6G	12A1	12A5
	Size	1	9	18	54	2	2	4	4	4	4	4	8	8	8	8	8	8	8	54	18	18	18	18	36	36	36	36	108	54	54
	2 P	1A	1A	1A	1A	3A	3B	3C	3D	3E	3F-1	3F1	3G	3H	3I	3J	3K	3L-1	3L1	2A	3A	3B	3A	3A	3D	3E	3F1	3F-1	3C	6B	6B
	3 P	1A	2A	2B	2C	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	4A	2A	2A	2B	2B	2B	2A	2B	2B	2C	4A	4A
	Type
648.723.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
648.723.1b	R	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$
648.723.1c	R	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$
648.723.1d	R	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$
648.723.2a	R	$2$	$2$	$0$	$0$	$2$	$- 1$	$2$	$2$	$- 1$	$2$	$2$	$- 1$	$2$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$2$	$0$	$2$	$0$	$- 1$	$0$	$- 1$	$0$	$0$	$0$	$- 1$	$- 1$
648.723.2b	R	$2$	$2$	$2$	$0$	$- 1$	$2$	$2$	$2$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$2$	$- 1$	$2$	$- 1$	$0$	$- 1$	$- 1$	$- 1$	$2$	$2$	$- 1$	$- 1$	$- 1$	$0$	$0$	$0$
648.723.2c	R	$2$	$- 2$	$0$	$0$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$2$	$0$	$0$	$- 2$	$0$	$- 2$	$0$	$- 2$	$0$	$0$	$0$	$0$	$0$
648.723.2d	R	$2$	$2$	$- 2$	$0$	$- 1$	$2$	$2$	$2$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$2$	$- 1$	$2$	$- 1$	$0$	$1$	$- 1$	$1$	$2$	$- 2$	$- 1$	$1$	$1$	$0$	$0$	$0$
648.723.2e	R	$2$	$2$	$0$	$0$	$2$	$- 1$	$2$	$2$	$- 1$	$2$	$2$	$- 1$	$2$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 2$	$0$	$2$	$0$	$- 1$	$0$	$- 1$	$0$	$0$	$0$	$1$	$1$
648.723.2f1	R	$2$	$- 2$	$0$	$0$	$2$	$- 1$	$2$	$2$	$- 1$	$2$	$2$	$- 1$	$2$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$0$	$0$	$- 2$	$0$	$1$	$0$	$1$	$0$	$0$	$0$	$- ζ_{12}^{- 1} - ζ_{12}$	$ζ_{12}^{- 1} + ζ_{12}$
648.723.2f2	R	$2$	$- 2$	$0$	$0$	$2$	$- 1$	$2$	$2$	$- 1$	$2$	$2$	$- 1$	$2$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$0$	$0$	$- 2$	$0$	$1$	$0$	$1$	$0$	$0$	$0$	$ζ_{12}^{- 1} + ζ_{12}$	$- ζ_{12}^{- 1} - ζ_{12}$
648.723.2g1	C	$2$	$- 2$	$0$	$0$	$- 1$	$2$	$2$	$2$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$2$	$- 1$	$2$	$- 1$	$0$	$- 1 - 2 ζ_{3}$	$1$	$1 + 2 ζ_{3}$	$- 2$	$0$	$1$	$1 + 2 ζ_{3}$	$- 1 - 2 ζ_{3}$	$0$	$0$	$0$
648.723.2g2	C	$2$	$- 2$	$0$	$0$	$- 1$	$2$	$2$	$2$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$2$	$- 1$	$2$	$- 1$	$0$	$1 + 2 ζ_{3}$	$1$	$- 1 - 2 ζ_{3}$	$- 2$	$0$	$1$	$- 1 - 2 ζ_{3}$	$1 + 2 ζ_{3}$	$0$	$0$	$0$
648.723.4a	R	$4$	$4$	$0$	$0$	$- 2$	$- 2$	$4$	$4$	$1$	$- 2$	$- 2$	$1$	$- 2$	$1$	$- 2$	$1$	$- 2$	$1$	$0$	$0$	$- 2$	$0$	$- 2$	$0$	$1$	$0$	$0$	$0$	$0$	$0$
648.723.4b	R	$4$	$0$	$0$	$2$	$4$	$4$	$1$	$- 2$	$4$	$- 2$	$- 2$	$1$	$1$	$- 2$	$1$	$- 2$	$- 2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 1$	$0$	$0$
648.723.4c	R	$4$	$0$	$2$	$0$	$4$	$4$	$- 2$	$1$	$4$	$1$	$1$	$- 2$	$- 2$	$1$	$- 2$	$1$	$1$	$- 2$	$0$	$2$	$0$	$2$	$0$	$- 1$	$0$	$- 1$	$- 1$	$0$	$0$	$0$
648.723.4d	R	$4$	$0$	$- 2$	$0$	$4$	$4$	$- 2$	$1$	$4$	$1$	$1$	$- 2$	$- 2$	$1$	$- 2$	$1$	$1$	$- 2$	$0$	$- 2$	$0$	$- 2$	$0$	$1$	$0$	$1$	$1$	$0$	$0$	$0$
648.723.4e	R	$4$	$0$	$0$	$- 2$	$4$	$4$	$1$	$- 2$	$4$	$- 2$	$- 2$	$1$	$1$	$- 2$	$1$	$- 2$	$- 2$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$1$	$0$	$0$
648.723.4f	R	$4$	$- 4$	$0$	$0$	$- 2$	$- 2$	$4$	$4$	$1$	$- 2$	$- 2$	$1$	$- 2$	$1$	$- 2$	$1$	$- 2$	$1$	$0$	$0$	$2$	$0$	$2$	$0$	$- 1$	$0$	$0$	$0$	$0$	$0$
648.723.4g1	C	$4$	$0$	$2$	$0$	$- 2$	$4$	$- 2$	$1$	$- 2$	$- 2 - 3 ζ_{3}$	$1 + 3 ζ_{3}$	$1$	$1$	$1 + 3 ζ_{3}$	$- 2$	$- 2 - 3 ζ_{3}$	$1$	$1$	$0$	$2 ζ_{3}$	$0$	$2 ζ_{3}^{- 1}$	$0$	$- 1$	$0$	$- ζ_{3}^{- 1}$	$- ζ_{3}$	$0$	$0$	$0$
648.723.4g2	C	$4$	$0$	$2$	$0$	$- 2$	$4$	$- 2$	$1$	$- 2$	$1 + 3 ζ_{3}$	$- 2 - 3 ζ_{3}$	$1$	$1$	$- 2 - 3 ζ_{3}$	$- 2$	$1 + 3 ζ_{3}$	$1$	$1$	$0$	$2 ζ_{3}^{- 1}$	$0$	$2 ζ_{3}$	$0$	$- 1$	$0$	$- ζ_{3}$	$- ζ_{3}^{- 1}$	$0$	$0$	$0$
648.723.4h1	C	$4$	$0$	$- 2$	$0$	$- 2$	$4$	$- 2$	$1$	$- 2$	$- 2 - 3 ζ_{3}$	$1 + 3 ζ_{3}$	$1$	$1$	$1 + 3 ζ_{3}$	$- 2$	$- 2 - 3 ζ_{3}$	$1$	$1$	$0$	$- 2 ζ_{3}$	$0$	$- 2 ζ_{3}^{- 1}$	$0$	$1$	$0$	$ζ_{3}^{- 1}$	$ζ_{3}$	$0$	$0$	$0$
648.723.4h2	C	$4$	$0$	$- 2$	$0$	$- 2$	$4$	$- 2$	$1$	$- 2$	$1 + 3 ζ_{3}$	$- 2 - 3 ζ_{3}$	$1$	$1$	$- 2 - 3 ζ_{3}$	$- 2$	$1 + 3 ζ_{3}$	$1$	$1$	$0$	$- 2 ζ_{3}^{- 1}$	$0$	$- 2 ζ_{3}$	$0$	$1$	$0$	$ζ_{3}$	$ζ_{3}^{- 1}$	$0$	$0$	$0$
648.723.8a	R	$8$	$0$	$0$	$0$	$8$	$- 4$	$- 4$	$2$	$- 4$	$2$	$2$	$2$	$- 4$	$- 1$	$2$	$- 1$	$- 1$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
648.723.8b	R	$8$	$0$	$0$	$0$	$8$	$- 4$	$2$	$- 4$	$- 4$	$- 4$	$- 4$	$- 1$	$2$	$2$	$- 1$	$2$	$2$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
648.723.8c	R	$8$	$0$	$0$	$0$	$- 4$	$8$	$2$	$- 4$	$- 4$	$2$	$2$	$- 1$	$- 1$	$2$	$2$	$2$	$- 4$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
648.723.8d	R	$8$	$0$	$0$	$0$	$- 4$	$- 4$	$2$	$- 4$	$2$	$2$	$2$	$- 4$	$- 1$	$- 1$	$- 1$	$- 1$	$2$	$5$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
648.723.8e	R	$8$	$0$	$0$	$0$	$- 4$	$- 4$	$2$	$- 4$	$2$	$2$	$2$	$5$	$- 1$	$- 1$	$- 1$	$- 1$	$2$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
648.723.8f1	C	$8$	$0$	$0$	$0$	$- 4$	$- 4$	$- 4$	$2$	$2$	$- 4 - 6 ζ_{3}$	$2 + 6 ζ_{3}$	$- 1$	$2$	$- 1 - 3 ζ_{3}$	$2$	$2 + 3 ζ_{3}$	$- 1$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
648.723.8f2	C	$8$	$0$	$0$	$0$	$- 4$	$- 4$	$- 4$	$2$	$2$	$2 + 6 ζ_{3}$	$- 4 - 6 ζ_{3}$	$- 1$	$2$	$2 + 3 ζ_{3}$	$2$	$- 1 - 3 ζ_{3}$	$- 1$	$- 1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

magma:CharacterTable(G);

Regular extensions

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Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
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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	36T1160

Degree	$36$
Order	$648$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$C_3^3:D_{12}$