Galois group: 32T418

• Label: 32T418
• Degree: $32$
• Order: $96$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• pp-group: no
• Group: $C_2^3:A_4$

Group invariants

Abstract group:		$C_2^3:A_4$
Order:		$96=2^{5} \cdot 3$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$32$
Transitive number $t$:		$418$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,26,13,2,25,14)(3,30,11,4,29,12)(5,32,15,6,31,16)(7,28,9,8,27,10)(17,18)(19,22,23,20,21,24)$, $(1,12)(2,11)(3,10)(4,9)(5,16)(6,15)(7,14)(8,13)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)$, $(1,23)(2,24)(3,21)(4,22)(5,19)(6,20)(7,17)(8,18)(9,27)(10,28)(11,25)(12,26)(13,31)(14,32)(15,29)(16,30)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$3$:	$C_3$
$6$:	$C_6$
$12$:	$A_4$ x 5
$24$:	$A_4\times C_2$ x 5
$48$:	$C_2^4:C_3$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $A_4$ x 5

Degree 8: $A_4\times C_2$ x 5

Degree 16: 16T64

Low degree siblings

12T56 x 30, 24T176 x 5, 24T177 x 30, 24T178 x 10

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	$2^{16}$	$1$	$2$	$16$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
2B	$2^{16}$	$3$	$2$	$16$	$( 1,15)( 2,16)( 3,13)( 4,14)( 5,11)( 6,12)( 7, 9)( 8,10)(17,27)(18,28)(19,25)(20,26)(21,31)(22,32)(23,29)(24,30)$
2C	$2^{16}$	$3$	$2$	$16$	$( 1, 6)( 2, 5)( 3, 8)( 4, 7)( 9,14)(10,13)(11,16)(12,15)(17,22)(18,21)(19,24)(20,23)(25,30)(26,29)(27,32)(28,31)$
2D	$2^{16}$	$3$	$2$	$16$	$( 1,12)( 2,11)( 3,10)( 4, 9)( 5,16)( 6,15)( 7,14)( 8,13)(17,32)(18,31)(19,30)(20,29)(21,28)(22,27)(23,26)(24,25)$
2E	$2^{16}$	$3$	$2$	$16$	$( 1,26)( 2,25)( 3,28)( 4,27)( 5,30)( 6,29)( 7,32)( 8,31)( 9,22)(10,21)(11,24)(12,23)(13,18)(14,17)(15,20)(16,19)$
2F	$2^{16}$	$3$	$2$	$16$	$( 1,20)( 2,19)( 3,18)( 4,17)( 5,24)( 6,23)( 7,22)( 8,21)( 9,32)(10,31)(11,30)(12,29)(13,28)(14,27)(15,26)(16,25)$
2G	$2^{16}$	$3$	$2$	$16$	$( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,15)(10,16)(11,13)(12,14)(17,23)(18,24)(19,21)(20,22)(25,31)(26,32)(27,29)(28,30)$
2H	$2^{16}$	$3$	$2$	$16$	$( 1, 9)( 2,10)( 3,11)( 4,12)( 5,13)( 6,14)( 7,15)( 8,16)(17,29)(18,30)(19,31)(20,32)(21,25)(22,26)(23,27)(24,28)$
2I	$2^{16}$	$3$	$2$	$16$	$( 1,27)( 2,28)( 3,25)( 4,26)( 5,31)( 6,32)( 7,29)( 8,30)( 9,23)(10,24)(11,21)(12,22)(13,19)(14,20)(15,17)(16,18)$
2J	$2^{16}$	$3$	$2$	$16$	$( 1,17)( 2,18)( 3,19)( 4,20)( 5,21)( 6,22)( 7,23)( 8,24)( 9,29)(10,30)(11,31)(12,32)(13,25)(14,26)(15,27)(16,28)$
2K	$2^{16}$	$3$	$2$	$16$	$( 1,16)( 2,15)( 3,14)( 4,13)( 5,12)( 6,11)( 7,10)( 8, 9)(17,28)(18,27)(19,26)(20,25)(21,32)(22,31)(23,30)(24,29)$
3A1	$3^{10},1^{2}$	$16$	$3$	$20$	$( 1,21,27)( 2,22,28)( 3,17,29)( 4,18,30)( 5,19,25)( 6,20,26)( 7,23,31)( 8,24,32)( 9,15,13)(10,16,14)$
3A-1	$3^{10},1^{2}$	$16$	$3$	$20$	$( 1,27,21)( 2,28,22)( 3,29,17)( 4,30,18)( 5,25,19)( 6,26,20)( 7,31,23)( 8,32,24)( 9,13,15)(10,14,16)$
6A1	$6^{5},2$	$16$	$6$	$26$	$( 1,28,21, 2,27,22)( 3,30,17, 4,29,18)( 5,26,19, 6,25,20)( 7,32,23, 8,31,24)( 9,14,15,10,13,16)(11,12)$
6A-1	$6^{5},2$	$16$	$6$	$26$	$( 1,22,27, 2,21,28)( 3,18,29, 4,17,30)( 5,20,25, 6,19,26)( 7,24,31, 8,23,32)( 9,16,13,10,15,14)(11,12)$

Malle's constant $a(G)$:     $1/16$

Character table

		1A	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	2K	3A1	3A-1	6A1	6A-1
	Size	1	1	3	3	3	3	3	3	3	3	3	3	16	16	16	16
	2 P	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	1A	3A-1	3A1	3A1	3A-1
	3 P	1A	2A	2B	2C	2D	2E	2F	2G	2H	2I	2J	2K	1A	1A	2A	2A
	Type
96.229.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
96.229.1b	R	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$-1$
96.229.1c1	C	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_3^{-1}$	${\zeta }_3$	${\zeta }_3$	${\zeta }_3^{-1}$
96.229.1c2	C	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	${\zeta }_3$	${\zeta }_3^{-1}$	${\zeta }_3^{-1}$	${\zeta }_3$
96.229.1d1	C	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	${\zeta }_3^{-1}$	${\zeta }_3$	$-{\zeta }_3$	$-{\zeta }_3^{-1}$
96.229.1d2	C	$1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$-1$	${\zeta }_3$	${\zeta }_3^{-1}$	$-{\zeta }_3^{-1}$	$-{\zeta }_3$
96.229.3a	R	$3$	$3$	$-1$	$-1$	$-1$	$-1$	$-1$	$3$	$-1$	$-1$	$-1$	$3$	$0$	$0$	$0$	$0$
96.229.3b	R	$3$	$3$	$-1$	$-1$	$-1$	$3$	$-1$	$-1$	$-1$	$-1$	$3$	$-1$	$0$	$0$	$0$	$0$
96.229.3c	R	$3$	$3$	$-1$	$-1$	$3$	$-1$	$-1$	$-1$	$-1$	$3$	$-1$	$-1$	$0$	$0$	$0$	$0$
96.229.3d	R	$3$	$3$	$-1$	$3$	$-1$	$-1$	$-1$	$-1$	$3$	$-1$	$-1$	$-1$	$0$	$0$	$0$	$0$
96.229.3e	R	$3$	$3$	$3$	$-1$	$-1$	$-1$	$3$	$-1$	$-1$	$-1$	$-1$	$-1$	$0$	$0$	$0$	$0$
96.229.3f	R	$3$	$-3$	$-1$	$-1$	$-1$	$-3$	$1$	$-1$	$1$	$1$	$3$	$1$	$0$	$0$	$0$	$0$
96.229.3g	R	$3$	$-3$	$-1$	$-1$	$-1$	$1$	$1$	$3$	$1$	$1$	$-1$	$-3$	$0$	$0$	$0$	$0$
96.229.3h	R	$3$	$-3$	$-1$	$-1$	$3$	$1$	$1$	$-1$	$1$	$-3$	$-1$	$1$	$0$	$0$	$0$	$0$
96.229.3i	R	$3$	$-3$	$-1$	$3$	$-1$	$1$	$1$	$-1$	$-3$	$1$	$-1$	$1$	$0$	$0$	$0$	$0$
96.229.3j	R	$3$	$-3$	$3$	$-1$	$-1$	$1$	$-3$	$-1$	$1$	$1$	$-1$	$1$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed