Galois group: 32T50

• Label: 32T50
• Degree: $32$
• Order: $32$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $Q_8:C_4$

Group invariants

Abstract group:		$Q_8:C_4$
Order:		$32=2^{5}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$3$

Group action invariants

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

Low degree resolvents

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

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 4

Degree 8: $C_4\times C_2$, $D_4$ x 2, $QD_{16}$, $C_2^2:C_4$ x 2

Degree 16: $C_2^2 : C_4$, $QD_{16}$, $Q_{16}$

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	$2^{16}$	$1$	$2$	$16$	$( 1,26)( 2,25)( 3,27)( 4,28)( 5,16)( 6,15)( 7,13)( 8,14)( 9,18)(10,17)(11,20)(12,19)(21,31)(22,32)(23,30)(24,29)$
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	$2^{16}$	$1$	$2$	$16$	$( 1,25)( 2,26)( 3,28)( 4,27)( 5,15)( 6,16)( 7,14)( 8,13)( 9,17)(10,18)(11,19)(12,20)(21,32)(22,31)(23,29)(24,30)$
4A	$4^{8}$	$2$	$4$	$24$	$( 1,19, 2,20)( 3,17, 4,18)( 5,24, 6,23)( 7,22, 8,21)( 9,27,10,28)(11,26,12,25)(13,32,14,31)(15,30,16,29)$
4B	$4^{8}$	$2$	$4$	$24$	$( 1,11, 2,12)( 3, 9, 4,10)( 5,30, 6,29)( 7,31, 8,32)(13,21,14,22)(15,24,16,23)(17,27,18,28)(19,26,20,25)$
4C	$4^{8}$	$4$	$4$	$24$	$( 1,10, 2, 9)( 3,11, 4,12)( 5,14, 6,13)( 7,16, 8,15)(17,25,18,26)(19,27,20,28)(21,29,22,30)(23,31,24,32)$
4D	$4^{8}$	$4$	$4$	$24$	$( 1, 3, 2, 4)( 5,22, 6,21)( 7,23, 8,24)( 9,11,10,12)(13,30,14,29)(15,31,16,32)(17,19,18,20)(25,28,26,27)$
4E1	$4^{8}$	$4$	$4$	$24$	$( 1,14,25, 7)( 2,13,26, 8)( 3,15,28, 5)( 4,16,27, 6)( 9,23,17,29)(10,24,18,30)(11,21,19,32)(12,22,20,31)$
4E-1	$4^{8}$	$4$	$4$	$24$	$( 1,22,25,31)( 2,21,26,32)( 3,24,28,30)( 4,23,27,29)( 5, 9,15,17)( 6,10,16,18)( 7,11,14,19)( 8,12,13,20)$
8A1	$8^{4}$	$2$	$8$	$28$	$( 1,23,11,15, 2,24,12,16)( 3,22, 9,13, 4,21,10,14)( 5,26,30,20, 6,25,29,19)( 7,28,31,17, 8,27,32,18)$
8A-1	$8^{4}$	$2$	$8$	$28$	$( 1,16,12,24, 2,15,11,23)( 3,14,10,21, 4,13, 9,22)( 5,19,29,25, 6,20,30,26)( 7,18,32,27, 8,17,31,28)$
8A3	$8^{4}$	$2$	$8$	$28$	$( 1,15,12,23, 2,16,11,24)( 3,13,10,22, 4,14, 9,21)( 5,20,29,26, 6,19,30,25)( 7,17,32,28, 8,18,31,27)$
8A-3	$8^{4}$	$2$	$8$	$28$	$( 1,24,11,16, 2,23,12,15)( 3,21, 9,14, 4,22,10,13)( 5,25,30,19, 6,26,29,20)( 7,27,31,18, 8,28,32,17)$

Malle's constant $a(G)$:     $1/16$

Character table

		1A	2A	2B	2C	4A	4B	4C	4D	4E1	4E-1	8A1	8A-1	8A3	8A-3
	Size	1	1	1	1	2	2	4	4	4	4	2	2	2	2
	2 P	1A	1A	1A	1A	2B	2B	2B	2B	2C	2C	4B	4B	4B	4B
	Type
32.10.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
32.10.1b	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$
32.10.1c	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$-1$
32.10.1d	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$
32.10.1e1	C	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-i$	$-1$	$i$	$1$	$-i$	$-i$	$i$	$i$
32.10.1e2	C	$1$	$-1$	$1$	$-1$	$1$	$-1$	$i$	$-1$	$-i$	$1$	$i$	$i$	$-i$	$-i$
32.10.1f1	C	$1$	$-1$	$1$	$-1$	$1$	$-1$	$-i$	$1$	$i$	$-1$	$i$	$i$	$-i$	$-i$
32.10.1f2	C	$1$	$-1$	$1$	$-1$	$1$	$-1$	$i$	$1$	$-i$	$-1$	$-i$	$-i$	$i$	$i$
32.10.2a	R	$2$	$-2$	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
32.10.2b	R	$2$	$2$	$2$	$2$	$-2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
32.10.2c1	C	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	$-{\zeta }_8-{\zeta }_8^3$	${\zeta }_8+{\zeta }_8^3$	${\zeta }_8+{\zeta }_8^3$	$-{\zeta }_8-{\zeta }_8^3$
32.10.2c2	C	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_8+{\zeta }_8^3$	$-{\zeta }_8-{\zeta }_8^3$	$-{\zeta }_8-{\zeta }_8^3$	${\zeta }_8+{\zeta }_8^3$
32.10.2d1	S	$2$	$2$	$-2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	$-{\zeta }_8^{-1}-{\zeta }_8$	${\zeta }_8^{-1}+{\zeta }_8$	$-{\zeta }_8^{-1}-{\zeta }_8$	${\zeta }_8^{-1}+{\zeta }_8$
32.10.2d2	S	$2$	$2$	$-2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$	${\zeta }_8^{-1}+{\zeta }_8$	$-{\zeta }_8^{-1}-{\zeta }_8$	${\zeta }_8^{-1}+{\zeta }_8$	$-{\zeta }_8^{-1}-{\zeta }_8$

Regular extensions

Data not computed