Galois group: 36T21

• Label: 36T21
• Degree: $36$
• Order: $72$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• pp-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 $t$:		$21$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$12$
Generators:		$(1,13,27,3,16,25)(2,14,28,4,15,26)(5,9,31,33,19,21)(6,10,32,34,20,22)(7,12,29,36,17,23)(8,11,30,35,18,24)$, $(1,34,7)(2,33,8)(3,36,6)(4,35,5)(9,17,15,10,18,16)(11,20,14,12,19,13)(21,29,28,22,30,27)(23,31,25,24,32,26)$

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: $C_2$

Degree 3: $C_3$, $S_3$

Degree 4: None

Degree 6: $C_6$, $S_3$, $A_4$, $S_3\times C_3$, $A_4\times C_2$

Degree 9: $S_3\times C_3$

Degree 12: $A_4 \times C_2$

Degree 18: $S_3 \times C_3$, 18T31, 18T32

Low degree siblings

12T43, 18T31, 18T32, 24T78, 24T83, 36T50, 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	$2^{12},1^{12}$	$3$	$2$	$12$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)$
2B	$2^{18}$	$3$	$2$	$18$	$( 1, 6)( 2, 5)( 3, 7)( 4, 8)( 9,11)(10,12)(13,17)(14,18)(15,19)(16,20)(21,24)(22,23)(25,29)(26,30)(27,32)(28,31)(33,35)(34,36)$
2C	$2^{18}$	$9$	$2$	$18$	$( 1, 4)( 2, 3)( 5,34)( 6,33)( 7,35)( 8,36)( 9,20)(10,19)(11,17)(12,18)(13,15)(14,16)(21,31)(22,32)(23,29)(24,30)(25,27)(26,28)$
3A	$3^{12}$	$2$	$3$	$24$	$( 1, 7,34)( 2, 8,33)( 3, 6,36)( 4, 5,35)( 9,15,18)(10,16,17)(11,14,19)(12,13,20)(21,28,30)(22,27,29)(23,25,32)(24,26,31)$
3B1	$3^{12}$	$4$	$3$	$24$	$( 1,16,28)( 2,15,27)( 3,13,26)( 4,14,25)( 5,19,32)( 6,20,31)( 7,17,30)( 8,18,29)( 9,22,33)(10,21,34)(11,23,35)(12,24,36)$
3B-1	$3^{12}$	$4$	$3$	$24$	$( 1,28,16)( 2,27,15)( 3,26,13)( 4,25,14)( 5,32,19)( 6,31,20)( 7,30,17)( 8,29,18)( 9,33,22)(10,34,21)(11,35,23)(12,36,24)$
3C1	$3^{12}$	$8$	$3$	$24$	$( 1,29,10)( 2,30, 9)( 3,32,12)( 4,31,11)( 5,24,14)( 6,23,13)( 7,22,16)( 8,21,15)(17,34,27)(18,33,28)(19,35,26)(20,36,25)$
3C-1	$3^{12}$	$8$	$3$	$24$	$( 1,17,22)( 2,18,21)( 3,20,23)( 4,19,24)( 5,11,26)( 6,12,25)( 7,10,27)( 8, 9,28)(13,32,36)(14,31,35)(15,30,33)(16,29,34)$
6A	$6^{4},3^{4}$	$6$	$6$	$28$	$( 1,33, 7, 2,34, 8)( 3,35, 6, 4,36, 5)( 9,18,15)(10,17,16)(11,19,14)(12,20,13)(21,29,28,22,30,27)(23,31,25,24,32,26)$
6B1	$6^{6}$	$12$	$6$	$30$	$( 1,31,16, 6,28,20)( 2,32,15, 5,27,19)( 3,30,13, 7,26,17)( 4,29,14, 8,25,18)( 9,35,22,11,33,23)(10,36,21,12,34,24)$
6B-1	$6^{6}$	$12$	$6$	$30$	$( 1,20,28, 6,16,31)( 2,19,27, 5,15,32)( 3,17,26, 7,13,30)( 4,18,25, 8,14,29)( 9,23,33,11,22,35)(10,24,34,12,21,36)$

Malle's constant $a(G)$:     $1/12$

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