Galois group: 32T2144

• Label: 32T2144
• Degree: $32$
• Order: $192$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• pp-group: no
• Group: $C_2^3.S_4$

Group invariants

Abstract group:		$C_2^3.S_4$
Order:		$192=2^{6} \cdot 3$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$32$
Transitive number $t$:		$2144$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,15,18,27)(2,16,17,28)(3,10,20,30)(4,9,19,29)(5,12,22,31)(6,11,21,32)(7,13,23,25)(8,14,24,26)$, $(1,9,30,20)(2,10,29,19)(3,13,31,24)(4,14,32,23)(5,12,25,17)(6,11,26,18)(7,16,27,22)(8,15,28,21)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$
$6$:	$S_3$
$12$:	$C_3 : C_4$
$24$:	$S_4$
$48$:	12T27

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $S_4$

Degree 8: $S_4$

Degree 16: 16T430

Low degree siblings

16T430, 24T374, 24T378, 24T384, 24T387

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	$1^{32}$	$1$	$1$	$0$	$()$
2A	$2^{16}$	$3$	$2$	$16$	$( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,23)(22,24)(25,28)(26,27)(29,31)(30,32)$
2B	$2^{12},1^{8}$	$4$	$2$	$12$	$( 1, 8)( 2, 7)( 3, 6)( 4, 5)( 9,14)(10,13)(11,16)(12,15)(17,20)(18,19)(21,23)(22,24)$
2C	$2^{16}$	$12$	$2$	$16$	$( 1,19)( 2,20)( 3,17)( 4,18)( 5,24)( 6,23)( 7,21)( 8,22)( 9,31)(10,32)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)$
3A	$3^{10},1^{2}$	$32$	$3$	$20$	$( 1,10,18)( 2, 9,17)( 3,15,21)( 4,16,22)( 5,11,24)( 6,12,23)( 7,14,20)( 8,13,19)(27,31,29)(28,32,30)$
4A	$4^{8}$	$12$	$4$	$24$	$( 1,13, 4,16)( 2,14, 3,15)( 5,10, 8,11)( 6, 9, 7,12)(17,27,20,26)(18,28,19,25)(21,31,23,29)(22,32,24,30)$
4B1	$4^{8}$	$24$	$4$	$24$	$( 1,27,19,14)( 2,28,20,13)( 3,30,17,11)( 4,29,18,12)( 5,31,24, 9)( 6,32,23,10)( 7,25,21,16)( 8,26,22,15)$
4B-1	$4^{8}$	$24$	$4$	$24$	$( 1,14,19,27)( 2,13,20,28)( 3,11,17,30)( 4,12,18,29)( 5, 9,24,31)( 6,10,23,32)( 7,16,21,25)( 8,15,22,26)$
4C1	$4^{6},2^{4}$	$24$	$4$	$22$	$( 1, 2)( 3, 5)( 4, 6)( 7, 8)( 9,24,12,19)(10,23,11,20)(13,21,16,17)(14,22,15,18)(25,27,32,29)(26,28,31,30)$
4C-1	$4^{6},2^{4}$	$24$	$4$	$22$	$( 1, 2)( 3, 5)( 4, 6)( 7, 8)( 9,19,12,24)(10,20,11,23)(13,17,16,21)(14,18,15,22)(25,29,32,27)(26,30,31,28)$
6A	$6^{4},3^{2},1^{2}$	$32$	$6$	$24$	$( 1,19,10, 8,18,13)( 2,20, 9, 7,17,14)( 3,23,15, 6,21,12)( 4,24,16, 5,22,11)(27,29,31)(28,30,32)$

Malle's constant $a(G)$:     $1/12$

Character table

		1A	2A	2B	2C	3A	4A	4B1	4B-1	4C1	4C-1	6A
	Size	1	3	4	12	32	12	24	24	24	24	32
	2 P	1A	1A	1A	1A	3A	2A	2C	2C	2B	2B	3A
	3 P	1A	2A	2B	2C	1A	4A	4B-1	4B1	4C-1	4C1	2B
	Type
192.185.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
192.185.1b	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$
192.185.1c1	C	$1$	$1$	$-1$	$-1$	$1$	$1$	$-i$	$i$	$-i$	$i$	$-1$
192.185.1c2	C	$1$	$1$	$-1$	$-1$	$1$	$1$	$i$	$-i$	$i$	$-i$	$-1$
192.185.2a	R	$2$	$2$	$2$	$2$	$-1$	$2$	$0$	$0$	$0$	$0$	$-1$
192.185.2b	S	$2$	$2$	$-2$	$-2$	$-1$	$2$	$0$	$0$	$0$	$0$	$1$
192.185.3a	R	$3$	$3$	$3$	$-1$	$0$	$-1$	$-1$	$-1$	$1$	$1$	$0$
192.185.3b	R	$3$	$3$	$3$	$-1$	$0$	$-1$	$1$	$1$	$-1$	$-1$	$0$
192.185.3c1	C	$3$	$3$	$-3$	$1$	$0$	$-1$	$-i$	$i$	$i$	$-i$	$0$
192.185.3c2	C	$3$	$3$	$-3$	$1$	$0$	$-1$	$i$	$-i$	$-i$	$i$	$0$
192.185.12a	R	$12$	$-4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed