LMFDB - Galois group: 36T1105

Show commands: Magma

magma:G := TransitiveGroup(36, 1105);

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 $t$:	$1105$	magma:t, n := TransitiveGroupIdentification(G); t;
Parity:	$-1$	magma:IsEven(G);
Primitive:	no	magma:IsPrimitive(G);
$\card{\Aut(F/K)}$:	$2$	magma:Order(Centralizer(SymmetricGroup(n), G));
Generators:	$(1,33,4,32,6,36,2,34,3,31,5,35)(7,27,15,20,11,30,13,22,9,25,18,24)(8,28,16,19,12,29,14,21,10,26,17,23)$, $(1,8,14,2,7,13)(3,12,16,5,9,18)(4,11,15,6,10,17)(19,23)(20,24)(25,36)(26,35)(27,33)(28,34)(29,32)(30,31)$	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: None
Degree 4: $D_{4}$
Degree 6: $S_3^2$, $C_3^2:D_4$
Degree 9: None
Degree 12: 12T36, 12T38, 12T169
Degree 18: 18T210

Low degree siblings

12T169 x 2, 18T210 x 2, 24T1521 x 2, 24T1533, 36T1105, 36T1106 x 2, 36T1107 x 2, 36T1153, 36T1154, 36T1160, 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	$2^{18}$	$9$	$2$	$18$	$( 1,13)( 2,14)( 3,15)( 4,16)( 5,17)( 6,18)( 7, 8)( 9,10)(11,12)(19,20)(21,22)(23,24)(25,32)(26,31)(27,34)(28,33)(29,36)(30,35)$
2B	$2^{18}$	$18$	$2$	$18$	$( 1,33)( 2,34)( 3,36)( 4,35)( 5,32)( 6,31)( 7,21)( 8,22)( 9,23)(10,24)(11,19)(12,20)(13,28)(14,27)(15,29)(16,30)(17,25)(18,26)$
2C	$2^{17},1^{2}$	$54$	$2$	$17$	$( 3, 6)( 4, 5)( 7,14)( 8,13)( 9,17)(10,18)(11,16)(12,15)(19,22)(20,21)(23,24)(25,28)(26,27)(29,30)(31,34)(32,33)(35,36)$
3A	$3^{12}$	$2$	$3$	$24$	$( 1, 3, 6)( 2, 4, 5)( 7, 9,11)( 8,10,12)(13,15,18)(14,16,17)(19,21,23)(20,22,24)(25,27,30)(26,28,29)(31,33,36)(32,34,35)$
3B	$3^{12}$	$2$	$3$	$24$	$( 1, 6, 3)( 2, 5, 4)( 7,11, 9)( 8,12,10)(13,18,15)(14,17,16)(19,21,23)(20,22,24)(25,27,30)(26,28,29)(31,33,36)(32,34,35)$
3C	$3^{6},1^{18}$	$4$	$3$	$12$	$(19,25,31)(20,26,32)(21,27,33)(22,28,34)(23,30,36)(24,29,35)$
3D	$3^{12}$	$4$	$3$	$24$	$( 1, 7,14)( 2, 8,13)( 3, 9,16)( 4,10,15)( 5,12,18)( 6,11,17)(19,31,25)(20,32,26)(21,33,27)(22,34,28)(23,36,30)(24,35,29)$
3E	$3^{6},1^{18}$	$4$	$3$	$12$	$( 1, 3, 6)( 2, 4, 5)( 7, 9,11)( 8,10,12)(13,15,18)(14,16,17)$
3F1	$3^{12}$	$4$	$3$	$24$	$( 1,17, 9)( 2,18,10)( 3,14,11)( 4,13,12)( 5,15, 8)( 6,16, 7)(19,30,33)(20,29,34)(21,25,36)(22,26,35)(23,27,31)(24,28,32)$
3F-1	$3^{12}$	$4$	$3$	$24$	$( 1, 9,17)( 2,10,18)( 3,11,14)( 4,12,13)( 5, 8,15)( 6, 7,16)(19,33,30)(20,34,29)(21,36,25)(22,35,26)(23,31,27)(24,32,28)$
3G	$3^{6},1^{18}$	$8$	$3$	$12$	$( 1,16,11)( 2,15,12)( 3,17, 7)( 4,18, 8)( 5,13,10)( 6,14, 9)$
3H	$3^{12}$	$8$	$3$	$24$	$( 1,16,11)( 2,15,12)( 3,17, 7)( 4,18, 8)( 5,13,10)( 6,14, 9)(19,30,33)(20,29,34)(21,25,36)(22,26,35)(23,27,31)(24,28,32)$
3I	$3^{12}$	$8$	$3$	$24$	$( 1, 6, 3)( 2, 5, 4)( 7,11, 9)( 8,12,10)(13,18,15)(14,17,16)(19,30,33)(20,29,34)(21,25,36)(22,26,35)(23,27,31)(24,28,32)$
3J	$3^{12}$	$8$	$3$	$24$	$( 1, 3, 6)( 2, 4, 5)( 7, 9,11)( 8,10,12)(13,15,18)(14,16,17)(19,30,33)(20,29,34)(21,25,36)(22,26,35)(23,27,31)(24,28,32)$
3K	$3^{12}$	$8$	$3$	$24$	$( 1,14, 7)( 2,13, 8)( 3,16, 9)( 4,15,10)( 5,18,12)( 6,17,11)(19,21,23)(20,22,24)(25,27,30)(26,28,29)(31,33,36)(32,34,35)$
3L1	$3^{12}$	$8$	$3$	$24$	$( 1,14, 7)( 2,13, 8)( 3,16, 9)( 4,15,10)( 5,18,12)( 6,17,11)(19,30,33)(20,29,34)(21,25,36)(22,26,35)(23,27,31)(24,28,32)$
3L-1	$3^{12}$	$8$	$3$	$24$	$( 1, 7,14)( 2, 8,13)( 3, 9,16)( 4,10,15)( 5,12,18)( 6,11,17)(19,30,33)(20,29,34)(21,25,36)(22,26,35)(23,27,31)(24,28,32)$
4A	$4^{9}$	$54$	$4$	$27$	$( 1,24, 8,30)( 2,23, 7,29)( 3,22,10,27)( 4,21, 9,28)( 5,19,11,26)( 6,20,12,25)(13,36,14,35)(15,33,16,34)(17,32,18,31)$
6A	$6^{6}$	$18$	$6$	$30$	$( 1,18, 3,13, 6,15)( 2,17, 4,14, 5,16)( 7,12, 9, 8,11,10)(19,24,21,20,23,22)(25,35,27,32,30,34)(26,36,28,31,29,33)$
6B	$6^{6}$	$18$	$6$	$30$	$( 1,10, 6, 8, 3,12)( 2, 9, 5, 7, 4,11)(13,16,18,14,15,17)(19,29,21,26,23,28)(20,30,22,25,24,27)(31,35,33,32,36,34)$
6C1	$6^{6}$	$18$	$6$	$30$	$( 1,31, 3,33, 6,36)( 2,32, 4,34, 5,35)( 7,19, 9,21,11,23)( 8,20,10,22,12,24)(13,26,15,28,18,29)(14,25,16,27,17,30)$
6C-1	$6^{6}$	$18$	$6$	$30$	$( 1,33, 6,31, 3,36)( 2,34, 5,32, 4,35)( 7,21,11,19, 9,23)( 8,22,12,20,10,24)(13,28,18,26,15,29)(14,27,17,25,16,30)$
6D	$6^{6}$	$36$	$6$	$30$	$( 1,28, 7,22,14,34)( 2,27, 8,21,13,33)( 3,29, 9,24,16,35)( 4,30,10,23,15,36)( 5,25,12,19,18,31)( 6,26,11,20,17,32)$
6E	$6^{3},2^{9}$	$36$	$6$	$24$	$( 1, 5, 3, 2, 6, 4)( 7,18, 9,13,11,15)( 8,17,10,14,12,16)(19,20)(21,22)(23,24)(25,32)(26,31)(27,34)(28,33)(29,36)(30,35)$
6F1	$6^{6}$	$36$	$6$	$30$	$( 1,29,17,34, 9,20)( 2,30,18,33,10,19)( 3,26,14,35,11,22)( 4,25,13,36,12,21)( 5,27,15,31, 8,23)( 6,28,16,32, 7,24)$
6F-1	$6^{6}$	$36$	$6$	$30$	$( 1,20, 9,34,17,29)( 2,19,10,33,18,30)( 3,22,11,35,14,26)( 4,21,12,36,13,25)( 5,23, 8,31,15,27)( 6,24, 7,32,16,28)$
6G	$6^{3},2^{8},1^{2}$	$108$	$6$	$23$	$( 3, 6)( 4, 5)( 7,14)( 8,13)( 9,17)(10,18)(11,16)(12,15)(19,34,25,22,31,28)(20,33,26,21,32,27)(23,35,30,24,36,29)$
12A1	$12^{3}$	$54$	$12$	$33$	$( 1,25,10,24, 6,27, 8,20, 3,30,12,22)( 2,26, 9,23, 5,28, 7,19, 4,29,11,21)(13,32,16,36,18,34,14,31,15,35,17,33)$
12A5	$12^{3}$	$54$	$12$	$33$	$( 1,27,12,24, 3,25, 8,22, 6,30,10,20)( 2,28,11,23, 4,26, 7,21, 5,29, 9,19)(13,34,17,36,15,32,14,33,18,35,16,31)$

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	36T1105

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