Galois group: 32T48

• Label: 32T48
• Degree: $32$
• Order: $32$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• pp-group: yes
• Group: $C_8:C_4$

Group action invariants

Degree $n$:		$32$
Transitive number $t$:		$48$
Group:		$C_8:C_4$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$32$
Generators:		(1,30,11,6)(2,29,12,5)(3,32,9,8)(4,31,10,7)(13,28,21,18)(14,27,22,17)(15,25,23,19)(16,26,24,20), (1,18,3,20,2,17,4,19)(5,24,8,21,6,23,7,22)(9,26,12,27,10,25,11,28)(13,30,15,31,14,29,16,32)

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_4$ x 2, $C_2^2$
$8$:	$D_{4}$, $C_4\times C_2$, $Q_8$
$16$:	$D_{8}$, $C_4:C_4$, $Q_{16}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_4$ x 2, $C_2^2$, $D_{4}$ x 2

Degree 8: $C_4\times C_2$, $D_4$, $Q_8$, $D_{8}$ x 2

Degree 16: $C_4:C_4$, $D_{8}$, $Q_{16}$

Low degree siblings

There are no siblings 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	$2^{16}$	$1$	$2$	$16$	$( 1,12)( 2,11)( 3,10)( 4, 9)( 5,30)( 6,29)( 7,32)( 8,31)(13,22)(14,21)(15,24)(16,23)(17,28)(18,27)(19,26)(20,25)$
2B	$2^{16}$	$1$	$2$	$16$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
2C	$2^{16}$	$1$	$2$	$16$	$( 1,11)( 2,12)( 3, 9)( 4,10)( 5,29)( 6,30)( 7,31)( 8,32)(13,21)(14,22)(15,23)(16,24)(17,27)(18,28)(19,25)(20,26)$
4A	$4^{8}$	$2$	$4$	$24$	$( 1,10, 2, 9)( 3,11, 4,12)( 5,31, 6,32)( 7,30, 8,29)(13,24,14,23)(15,21,16,22)(17,26,18,25)(19,27,20,28)$
4B	$4^{8}$	$2$	$4$	$24$	$( 1, 4, 2, 3)( 5, 7, 6, 8)( 9,11,10,12)(13,16,14,15)(17,20,18,19)(21,24,22,23)(25,27,26,28)(29,31,30,32)$
4C1	$4^{8}$	$4$	$4$	$24$	$( 1,30,11, 6)( 2,29,12, 5)( 3,32, 9, 8)( 4,31,10, 7)(13,28,21,18)(14,27,22,17)(15,25,23,19)(16,26,24,20)$
4C-1	$4^{8}$	$4$	$4$	$24$	$( 1,15,11,23)( 2,16,12,24)( 3,13, 9,21)( 4,14,10,22)( 5,17,29,27)( 6,18,30,28)( 7,19,31,25)( 8,20,32,26)$
4D1	$4^{8}$	$4$	$4$	$24$	$( 1,23,11,15)( 2,24,12,16)( 3,21, 9,13)( 4,22,10,14)( 5,27,29,17)( 6,28,30,18)( 7,25,31,19)( 8,26,32,20)$
4D-1	$4^{8}$	$4$	$4$	$24$	$( 1, 6,11,30)( 2, 5,12,29)( 3, 8, 9,32)( 4, 7,10,31)(13,18,21,28)(14,17,22,27)(15,19,23,25)(16,20,24,26)$
8A1	$8^{4}$	$2$	$8$	$28$	$( 1,17, 3,19, 2,18, 4,20)( 5,23, 8,22, 6,24, 7,21)( 9,25,12,28,10,26,11,27)(13,29,15,32,14,30,16,31)$
8A3	$8^{4}$	$2$	$8$	$28$	$( 1,18, 3,20, 2,17, 4,19)( 5,24, 8,21, 6,23, 7,22)( 9,26,12,27,10,25,11,28)(13,30,15,31,14,29,16,32)$
8B1	$8^{4}$	$2$	$8$	$28$	$( 1,27, 3,25, 2,28, 4,26)( 5,15, 8,14, 6,16, 7,13)( 9,19,12,18,10,20,11,17)(21,29,23,32,22,30,24,31)$
8B3	$8^{4}$	$2$	$8$	$28$	$( 1,28, 3,26, 2,27, 4,25)( 5,16, 8,13, 6,15, 7,14)( 9,20,12,17,10,19,11,18)(21,30,23,31,22,29,24,32)$

Malle's constant $a(G)$:     $1/16$

Group invariants

Order:		$32=2^{5}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$3$
Label:		32.14
Character table:

		1A	2A	2B	2C	4A	4B	4C1	4C-1	4D1	4D-1	8A1	8A3	8B1	8B3
	Size	1	1	1	1	2	2	4	4	4	4	2	2	2	2
	2 P	1A	1A	1A	1A	2B	2B	2C	2C	2C	2C	4B	4B	4B	4B
	Type
32.14.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
32.14.1b	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$
32.14.1c	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$
32.14.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$-1$
32.14.1e1	C	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-i$	$i$	$-i$	$i$	$-1$	$-1$	$1$	$1$
32.14.1e2	C	$1$	$1$	$-1$	$-1$	$1$	$-1$	$i$	$-i$	$i$	$-i$	$-1$	$-1$	$1$	$1$
32.14.1f1	C	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-i$	$i$	$i$	$-i$	$1$	$1$	$-1$	$-1$
32.14.1f2	C	$1$	$1$	$-1$	$-1$	$1$	$-1$	$i$	$-i$	$-i$	$i$	$1$	$1$	$-1$	$-1$
32.14.2a	R	$2$	$2$	$2$	$2$	$-2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
32.14.2b	S	$2$	$2$	$-2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
32.14.2c1	R	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$-{\zeta }_8^{-1}-{\zeta }_8$	${\zeta }_8^{-1}+{\zeta }_8$	$-{\zeta }_8^{-1}-{\zeta }_8$	${\zeta }_8^{-1}+{\zeta }_8$
32.14.2c2	R	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_8^{-1}+{\zeta }_8$	$-{\zeta }_8^{-1}-{\zeta }_8$	${\zeta }_8^{-1}+{\zeta }_8$	$-{\zeta }_8^{-1}-{\zeta }_8$
32.14.2d1	S	$2$	$-2$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$-{\zeta }_8^{-1}-{\zeta }_8$	${\zeta }_8^{-1}+{\zeta }_8$	${\zeta }_8^{-1}+{\zeta }_8$	$-{\zeta }_8^{-1}-{\zeta }_8$
32.14.2d2	S	$2$	$-2$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_8^{-1}+{\zeta }_8$	$-{\zeta }_8^{-1}-{\zeta }_8$	$-{\zeta }_8^{-1}-{\zeta }_8$	${\zeta }_8^{-1}+{\zeta }_8$