LMFDB - Galois group: 36T150

Show commands: Magma

magma:G := TransitiveGroup(36, 150);

Group invariants

Abstract group:	$S_3\times D_{12}$	magma:IdentifyGroup(G);
Order:	$144=2^{4} \cdot 3^{2}$	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 $t$:	$150$	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:	$(1,10,26,36,13,21)(2,9,25,35,14,22)(3,11,27,34,16,24)(4,12,28,33,15,23)(5,18,31)(6,17,32)(7,19,29,8,20,30)$, $(1,34)(2,33)(3,35)(4,36)(5,7)(6,8)(9,16)(10,15)(11,13)(12,14)(17,19)(18,20)(21,28)(22,27)(23,25)(24,26)(29,31)(30,32)$, $(3,4)(5,34)(6,33)(7,36)(8,35)(9,30)(10,29)(11,31)(12,32)(13,26)(14,25)(15,27)(16,28)(17,23)(18,24)(19,22)(20,21)$	magma:Generators(G);

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$2$: $C_2$ x 7
$4$: $C_2^2$ x 7
$6$: $S_3$ x 2
$8$: $D_{4}$ x 2, $C_2^3$
$12$: $D_{6}$ x 6
$16$: $D_4\times C_2$
$24$: $S_3 \times C_2^2$ x 2, $D_{12}$ x 2
$36$: $S_3^2$
$48$: 12T28, 24T29
$72$: 12T37

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 7
$4$:	$C_2^2$ x 7
$6$:	$S_3$ x 2
$8$:	$D_{4}$ x 2, $C_2^3$
$12$:	$D_{6}$ x 6
$16$:	$D_4\times C_2$
$24$:	$S_3 \times C_2^2$ x 2, $D_{12}$ x 2
$36$:	$S_3^2$
$48$:	12T28, 24T29
$72$:	12T37

Subfields

Degree 2: $C_2$
Degree 3: $S_3$ x 2
Degree 4: $D_{4}$
Degree 6: $D_{6}$ x 2
Degree 9: $S_3^2$
Degree 12: $D_{12}$, 12T28
Degree 18: 18T29

Low degree siblings

24T229 x 2, 36T150 x 3
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	$2^{18}$	$1$	$2$	$18$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$
2B	$2^{12},1^{12}$	$3$	$2$	$12$	$( 9,22)(10,21)(11,24)(12,23)(13,26)(14,25)(15,28)(16,27)(17,32)(18,31)(19,30)(20,29)$
2C	$2^{18}$	$3$	$2$	$18$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,21)(10,22)(11,23)(12,24)(13,25)(14,26)(15,27)(16,28)(17,31)(18,32)(19,29)(20,30)(33,34)(35,36)$
2D	$2^{18}$	$6$	$2$	$18$	$( 1,34)( 2,33)( 3,35)( 4,36)( 5, 7)( 6, 8)( 9,16)(10,15)(11,13)(12,14)(17,19)(18,20)(21,28)(22,27)(23,25)(24,26)(29,31)(30,32)$
2E	$2^{15},1^{6}$	$6$	$2$	$15$	$( 1, 7)( 2, 8)( 3, 6)( 4, 5)( 9,10)(13,20)(14,19)(15,18)(16,17)(21,22)(25,30)(26,29)(27,32)(28,31)(35,36)$
2F	$2^{17},1^{2}$	$18$	$2$	$17$	$( 1,10)( 2, 9)( 3,11)( 4,12)( 5,18)( 6,17)( 7,19)( 8,20)(13,36)(14,35)(15,33)(16,34)(21,26)(22,25)(23,28)(24,27)(29,30)$
2G	$2^{18}$	$18$	$2$	$18$	$( 1,32)( 2,31)( 3,30)( 4,29)( 5,25)( 6,26)( 7,28)( 8,27)( 9,11)(10,12)(13,17)(14,18)(15,20)(16,19)(21,33)(22,34)(23,36)(24,35)$
3A	$3^{12}$	$2$	$3$	$24$	$( 1, 8,35)( 2, 7,36)( 3, 6,34)( 4, 5,33)( 9,13,19)(10,14,20)(11,16,17)(12,15,18)(21,25,29)(22,26,30)(23,28,31)(24,27,32)$
3B	$3^{12}$	$2$	$3$	$24$	$( 1,26,13)( 2,25,14)( 3,27,16)( 4,28,15)( 5,31,18)( 6,32,17)( 7,29,20)( 8,30,19)( 9,35,22)(10,36,21)(11,34,24)(12,33,23)$
3C	$3^{12}$	$4$	$3$	$24$	$( 1,22,19)( 2,21,20)( 3,24,17)( 4,23,18)( 5,28,12)( 6,27,11)( 7,25,10)( 8,26, 9)(13,35,30)(14,36,29)(15,33,31)(16,34,32)$
4A	$4^{9}$	$2$	$4$	$27$	$( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,12,10,11)(13,15,14,16)(17,19,18,20)(21,24,22,23)(25,27,26,28)(29,32,30,31)(33,36,34,35)$
4B	$4^{9}$	$6$	$4$	$27$	$( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,23,10,24)(11,22,12,21)(13,28,14,27)(15,25,16,26)(17,30,18,29)(19,31,20,32)(33,36,34,35)$
6A	$6^{6}$	$2$	$6$	$30$	$( 1, 7,35, 2, 8,36)( 3, 5,34, 4, 6,33)( 9,14,19,10,13,20)(11,15,17,12,16,18)(21,26,29,22,25,30)(23,27,31,24,28,32)$
6B	$6^{6}$	$2$	$6$	$30$	$( 1,14,26, 2,13,25)( 3,15,27, 4,16,28)( 5,17,31, 6,18,32)( 7,19,29, 8,20,30)( 9,21,35,10,22,36)(11,23,34,12,24,33)$
6C	$6^{6}$	$4$	$6$	$30$	$( 1,20,22, 2,19,21)( 3,18,24, 4,17,23)( 5,11,28, 6,12,27)( 7, 9,25, 8,10,26)(13,29,35,14,30,36)(15,32,33,16,31,34)$
6D	$6^{4},3^{4}$	$6$	$6$	$28$	$( 1,35, 8)( 2,36, 7)( 3,34, 6)( 4,33, 5)( 9,30,13,22,19,26)(10,29,14,21,20,25)(11,32,16,24,17,27)(12,31,15,23,18,28)$
6E	$6^{6}$	$6$	$6$	$30$	$( 1, 7,35, 2, 8,36)( 3, 5,34, 4, 6,33)( 9,25,19,21,13,29)(10,26,20,22,14,30)(11,28,17,23,16,31)(12,27,18,24,15,32)$
6F	$6^{6}$	$12$	$6$	$30$	$( 1,11,26,34,13,24)( 2,12,25,33,14,23)( 3, 9,27,35,16,22)( 4,10,28,36,15,21)( 5,20,31, 7,18,29)( 6,19,32, 8,17,30)$
6G	$6^{5},3^{2}$	$12$	$6$	$29$	$( 1,20,26, 7,13,29)( 2,19,25, 8,14,30)( 3,17,27, 6,16,32)( 4,18,28, 5,15,31)( 9,21,35,10,22,36)(11,24,34)(12,23,33)$
12A1	$12^{3}$	$2$	$12$	$33$	$( 1,34, 7, 4,35, 6, 2,33, 8, 3,36, 5)( 9,17,14,12,19,16,10,18,13,11,20,15)(21,31,26,24,29,28,22,32,25,23,30,27)$
12A5	$12^{3}$	$2$	$12$	$33$	$( 1, 6,36, 4, 8,34, 2, 5,35, 3, 7,33)( 9,16,20,12,13,17,10,15,19,11,14,18)(21,28,30,24,25,31,22,27,29,23,26,32)$
12B	$12^{3}$	$4$	$12$	$33$	$( 1,28,14, 3,26,15, 2,27,13, 4,25,16)( 5,29,17, 8,31,20, 6,30,18, 7,32,19)( 9,33,21,11,35,23,10,34,22,12,36,24)$
12C1	$12^{3}$	$4$	$12$	$33$	$( 1,24,20, 4,22,17, 2,23,19, 3,21,18)( 5,26,11, 7,28, 9, 6,25,12, 8,27,10)(13,34,29,15,35,32,14,33,30,16,36,31)$
12C5	$12^{3}$	$4$	$12$	$33$	$( 1,32,10, 4,30,11, 2,31, 9, 3,29,12)( 5,22,16, 7,23,13, 6,21,15, 8,24,14)(17,36,28,19,34,25,18,35,27,20,33,26)$
12D1	$12^{3}$	$6$	$12$	$33$	$( 1,34, 7, 4,35, 6, 2,33, 8, 3,36, 5)( 9,32,14,23,19,27,10,31,13,24,20,28)(11,29,15,22,17,25,12,30,16,21,18,26)$
12D5	$12^{3}$	$6$	$12$	$33$	$( 1, 6,36, 4, 8,34, 2, 5,35, 3, 7,33)( 9,27,20,23,13,32,10,28,19,24,14,31)(11,25,18,22,16,29,12,26,17,21,15,30)$

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

magma:ConjugacyClasses(G);

Character table

		1A	2A	2B	2C	2D	2E	2F	2G	3A	3B	3C	4A	4B	6A	6B	6C	6D	6E	6F	6G	12A1	12A5	12B	12C1	12C5	12D1	12D5
	Size	1	1	3	3	6	6	18	18	2	2	4	2	6	2	2	4	6	6	12	12	2	2	4	4	4	6	6
	2 P	1A	1A	1A	1A	1A	1A	1A	1A	3A	3B	3C	2A	2A	3A	3B	3C	3A	3A	3B	3B	6A	6A	6B	6C	6C	6A	6A
	3 P	1A	2A	2B	2C	2D	2E	2F	2G	1A	1A	1A	4A	4B	2A	2A	2A	2B	2C	2D	2E	4A	4A	4A	4A	4A	4B	4B
	Type
144.144.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
144.144.1b	R	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$
144.144.1c	R	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$
144.144.1d	R	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$- 1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$
144.144.1e	R	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$
144.144.1f	R	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
144.144.1g	R	$1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$
144.144.1h	R	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$
144.144.2a	R	$2$	$2$	$0$	$0$	$2$	$2$	$0$	$0$	$2$	$- 1$	$- 1$	$2$	$0$	$2$	$- 1$	$- 1$	$0$	$0$	$- 1$	$- 1$	$2$	$2$	$- 1$	$- 1$	$- 1$	$0$	$0$
144.144.2b	R	$2$	$2$	$2$	$2$	$0$	$0$	$0$	$0$	$- 1$	$2$	$- 1$	$2$	$2$	$- 1$	$2$	$- 1$	$- 1$	$- 1$	$0$	$0$	$- 1$	$- 1$	$2$	$- 1$	$- 1$	$- 1$	$- 1$
144.144.2c	R	$2$	$- 2$	$- 2$	$2$	$0$	$0$	$0$	$0$	$2$	$2$	$2$	$0$	$0$	$- 2$	$- 2$	$- 2$	$- 2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
144.144.2d	R	$2$	$- 2$	$2$	$- 2$	$0$	$0$	$0$	$0$	$2$	$2$	$2$	$0$	$0$	$- 2$	$- 2$	$- 2$	$2$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
144.144.2e	R	$2$	$2$	$- 2$	$- 2$	$0$	$0$	$0$	$0$	$- 1$	$2$	$- 1$	$- 2$	$2$	$- 1$	$2$	$- 1$	$1$	$1$	$0$	$0$	$1$	$1$	$- 2$	$1$	$1$	$- 1$	$- 1$
144.144.2f	R	$2$	$2$	$- 2$	$- 2$	$0$	$0$	$0$	$0$	$- 1$	$2$	$- 1$	$2$	$- 2$	$- 1$	$2$	$- 1$	$1$	$1$	$0$	$0$	$- 1$	$- 1$	$2$	$- 1$	$- 1$	$1$	$1$
144.144.2g	R	$2$	$2$	$0$	$0$	$- 2$	$- 2$	$0$	$0$	$2$	$- 1$	$- 1$	$2$	$0$	$2$	$- 1$	$- 1$	$0$	$0$	$1$	$1$	$2$	$2$	$- 1$	$- 1$	$- 1$	$0$	$0$
144.144.2h	R	$2$	$2$	$0$	$0$	$- 2$	$2$	$0$	$0$	$2$	$- 1$	$- 1$	$- 2$	$0$	$2$	$- 1$	$- 1$	$0$	$0$	$1$	$- 1$	$- 2$	$- 2$	$1$	$1$	$1$	$0$	$0$
144.144.2i	R	$2$	$2$	$0$	$0$	$2$	$- 2$	$0$	$0$	$2$	$- 1$	$- 1$	$- 2$	$0$	$2$	$- 1$	$- 1$	$0$	$0$	$- 1$	$1$	$- 2$	$- 2$	$1$	$1$	$1$	$0$	$0$
144.144.2j	R	$2$	$2$	$2$	$2$	$0$	$0$	$0$	$0$	$- 1$	$2$	$- 1$	$- 2$	$- 2$	$- 1$	$2$	$- 1$	$- 1$	$- 1$	$0$	$0$	$1$	$1$	$- 2$	$1$	$1$	$1$	$1$
144.144.2k1	R	$2$	$- 2$	$- 2$	$2$	$0$	$0$	$0$	$0$	$- 1$	$2$	$- 1$	$0$	$0$	$1$	$- 2$	$1$	$1$	$- 1$	$0$	$0$	$- ζ_{12}^{- 1} - ζ_{12}$	$ζ_{12}^{- 1} + ζ_{12}$	$0$	$- ζ_{12}^{- 1} - ζ_{12}$	$ζ_{12}^{- 1} + ζ_{12}$	$ζ_{12}^{- 1} + ζ_{12}$	$- ζ_{12}^{- 1} - ζ_{12}$
144.144.2k2	R	$2$	$- 2$	$- 2$	$2$	$0$	$0$	$0$	$0$	$- 1$	$2$	$- 1$	$0$	$0$	$1$	$- 2$	$1$	$1$	$- 1$	$0$	$0$	$ζ_{12}^{- 1} + ζ_{12}$	$- ζ_{12}^{- 1} - ζ_{12}$	$0$	$ζ_{12}^{- 1} + ζ_{12}$	$- ζ_{12}^{- 1} - ζ_{12}$	$- ζ_{12}^{- 1} - ζ_{12}$	$ζ_{12}^{- 1} + ζ_{12}$
144.144.2l1	R	$2$	$- 2$	$2$	$- 2$	$0$	$0$	$0$	$0$	$- 1$	$2$	$- 1$	$0$	$0$	$1$	$- 2$	$1$	$- 1$	$1$	$0$	$0$	$- ζ_{12}^{- 1} - ζ_{12}$	$ζ_{12}^{- 1} + ζ_{12}$	$0$	$- ζ_{12}^{- 1} - ζ_{12}$	$ζ_{12}^{- 1} + ζ_{12}$	$- ζ_{12}^{- 1} - ζ_{12}$	$ζ_{12}^{- 1} + ζ_{12}$
144.144.2l2	R	$2$	$- 2$	$2$	$- 2$	$0$	$0$	$0$	$0$	$- 1$	$2$	$- 1$	$0$	$0$	$1$	$- 2$	$1$	$- 1$	$1$	$0$	$0$	$ζ_{12}^{- 1} + ζ_{12}$	$- ζ_{12}^{- 1} - ζ_{12}$	$0$	$ζ_{12}^{- 1} + ζ_{12}$	$- ζ_{12}^{- 1} - ζ_{12}$	$ζ_{12}^{- 1} + ζ_{12}$	$- ζ_{12}^{- 1} - ζ_{12}$
144.144.4a	R	$4$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$- 2$	$- 2$	$1$	$4$	$0$	$- 2$	$- 2$	$1$	$0$	$0$	$0$	$0$	$- 2$	$- 2$	$- 2$	$1$	$1$	$0$	$0$
144.144.4b	R	$4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$4$	$- 2$	$- 2$	$0$	$0$	$- 4$	$2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
144.144.4c	R	$4$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$- 2$	$- 2$	$1$	$- 4$	$0$	$- 2$	$- 2$	$1$	$0$	$0$	$0$	$0$	$2$	$2$	$2$	$- 1$	$- 1$	$0$	$0$
144.144.4d1	R	$4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$- 2$	$- 2$	$1$	$0$	$0$	$2$	$2$	$- 1$	$0$	$0$	$0$	$0$	$- 2 ζ_{12}^{- 1} - 2 ζ_{12}$	$2 ζ_{12}^{- 1} + 2 ζ_{12}$	$0$	$ζ_{12}^{- 1} + ζ_{12}$	$- ζ_{12}^{- 1} - ζ_{12}$	$0$	$0$
144.144.4d2	R	$4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$- 2$	$- 2$	$1$	$0$	$0$	$2$	$2$	$- 1$	$0$	$0$	$0$	$0$	$2 ζ_{12}^{- 1} + 2 ζ_{12}$	$- 2 ζ_{12}^{- 1} - 2 ζ_{12}$	$0$	$- ζ_{12}^{- 1} - ζ_{12}$	$ζ_{12}^{- 1} + ζ_{12}$	$0$	$0$

magma:CharacterTable(G);

Regular extensions

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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	36T150

Degree	$36$
Order	$144$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$S_3\times D_{12}$