Galois group: 36T50

• Label: 36T50
• Degree: $36$
• Order: $72$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $S_3\times A_4$

Group invariants

Abstract group:		$S_3\times A_4$
Order:		$72=2^{3} \cdot 3^{2}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$36$
Transitive number $t$:		$50$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$4$
Generators:		$(1,10)(2,9)(3,12)(4,11)(13,18)(14,17)(15,20)(16,19)(21,24)(22,23)(25,32)(26,31)(27,30)(28,29)(33,36)(34,35)$, $(1,24,29,7,18,36)(2,23,30,8,17,35)(3,21,31,6,19,34)(4,22,32,5,20,33)(9,16,26,10,15,25)(11,13,28,12,14,27)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$6$:	$S_3$, $C_6$
$12$:	$A_4$
$18$:	$S_3\times C_3$
$24$:	$A_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 3: $C_3$, $S_3$

Degree 4: None

Degree 6: $A_4$, $A_4\times C_2$

Degree 9: $S_3\times C_3$

Degree 12: $A_4\times C_2$

Degree 18: 18T31, 18T32

Low degree siblings

12T43, 18T31, 18T32, 24T78, 24T83, 36T21, 36T51

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	$2^{18}$	$3$	$2$	$18$	$( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$
2B	$2^{18}$	$3$	$2$	$18$	$( 1, 2)( 3, 4)( 5,11)( 6,12)( 7,10)( 8, 9)(13,21)(14,22)(15,23)(16,24)(17,18)(19,20)(25,36)(26,35)(27,34)(28,33)(29,30)(31,32)$
2C	$2^{16},1^{4}$	$9$	$2$	$16$	$( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,12)(10,11)(17,24)(18,23)(19,22)(20,21)(25,27)(26,28)(29,33)(30,34)(31,35)(32,36)$
3A	$3^{12}$	$2$	$3$	$24$	$( 1,10, 8)( 2, 9, 7)( 3,12, 5)( 4,11, 6)(13,22,19)(14,21,20)(15,24,17)(16,23,18)(25,35,29)(26,36,30)(27,33,31)(28,34,32)$
3B1	$3^{12}$	$4$	$3$	$24$	$( 1,29,18)( 2,30,17)( 3,31,19)( 4,32,20)( 5,33,22)( 6,34,21)( 7,36,24)( 8,35,23)( 9,26,15)(10,25,16)(11,28,14)(12,27,13)$
3B-1	$3^{12}$	$4$	$3$	$24$	$( 1,18,29)( 2,17,30)( 3,19,31)( 4,20,32)( 5,22,33)( 6,21,34)( 7,24,36)( 8,23,35)( 9,15,26)(10,16,25)(11,14,28)(12,13,27)$
3C1	$3^{12}$	$8$	$3$	$24$	$( 1,14,36)( 2,13,35)( 3,15,34)( 4,16,33)( 5,17,28)( 6,18,27)( 7,19,25)( 8,20,26)( 9,22,29)(10,21,30)(11,23,31)(12,24,32)$
3C-1	$3^{12}$	$8$	$3$	$24$	$( 1,26,21)( 2,25,22)( 3,28,24)( 4,27,23)( 5,32,15)( 6,31,16)( 7,29,13)( 8,30,14)( 9,35,19)(10,36,20)(11,33,18)(12,34,17)$
6A	$6^{6}$	$6$	$6$	$30$	$( 1, 6,10, 4, 8,11)( 2, 5, 9, 3, 7,12)(13,18,22,16,19,23)(14,17,21,15,20,24)(25,30,35,26,29,36)(27,32,33,28,31,34)$
6B1	$6^{6}$	$12$	$6$	$30$	$( 1,17,29, 2,18,30)( 3,20,31, 4,19,32)( 5,14,33,11,22,28)( 6,13,34,12,21,27)( 7,16,36,10,24,25)( 8,15,35, 9,23,26)$
6B-1	$6^{6}$	$12$	$6$	$30$	$( 1,30,18, 2,29,17)( 3,32,19, 4,31,20)( 5,28,22,11,33,14)( 6,27,21,12,34,13)( 7,25,24,10,36,16)( 8,26,23, 9,35,15)$

Malle's constant $a(G)$:     $1/16$

Character table

		1A	2A	2B	2C	3A	3B1	3B-1	3C1	3C-1	6A	6B1	6B-1
	Size	1	3	3	9	2	4	4	8	8	6	12	12
	2 P	1A	1A	1A	1A	3A	3B-1	3B1	3C-1	3C1	3A	3B1	3B-1
	3 P	1A	2A	2B	2C	1A	1A	1A	1A	1A	2A	2B	2B
	Type
72.44.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
72.44.1b	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$
72.44.1c1	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$
72.44.1c2	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$
72.44.1d1	C	$1$	$1$	$-1$	$-1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$	$1$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$
72.44.1d2	C	$1$	$1$	$-1$	$-1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$	$1$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$
72.44.2a	R	$2$	$2$	$0$	$0$	$-1$	$2$	$2$	$-1$	$-1$	$-1$	$0$	$0$
72.44.2b1	C	$2$	$2$	$0$	$0$	$-1$	$2{\zeta }_3^{-1}$	$2{\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$	$-1$	$0$	$0$
72.44.2b2	C	$2$	$2$	$0$	$0$	$-1$	$2{\zeta }_3$	$2{\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$	$-1$	$0$	$0$
72.44.3a	R	$3$	$-1$	$3$	$-1$	$3$	$0$	$0$	$0$	$0$	$-1$	$0$	$0$
72.44.3b	R	$3$	$-1$	$-3$	$1$	$3$	$0$	$0$	$0$	$0$	$-1$	$0$	$0$
72.44.6a	R	$6$	$-2$	$0$	$0$	$-3$	$0$	$0$	$0$	$0$	$1$	$0$	$0$

Regular extensions

Data not computed