Galois group: 32T26

• Label: 32T26
• Degree: $32$
• Order: $32$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $D_8:C_2$

Group invariants

Abstract group:		$D_8:C_2$
Order:		$32=2^{5}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$3$

Group action invariants

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

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 7
$4$:	$C_2^2$ x 7
$8$:	$D_{4}$ x 2, $C_2^3$
$16$:	$D_4\times C_2$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 7

Degree 4: $C_2^2$ x 7, $D_{4}$ x 4

Degree 8: $C_2^3$, $D_4$ x 2, $D_4\times C_2$ x 4

Degree 16: $D_4\times C_2$, 16T44 x 2, 16T47

Low degree siblings

16T44 x 2, 16T47

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	$2^{16}$	$1$	$2$	$16$	$( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,12)(10,11)(13,15)(14,16)(17,20)(18,19)(21,23)(22,24)(25,28)(26,27)(29,31)(30,32)$
2B	$2^{16}$	$2$	$2$	$16$	$( 1,12)( 2,11)( 3,10)( 4, 9)( 5,30)( 6,29)( 7,32)( 8,31)(13,24)(14,23)(15,22)(16,21)(17,27)(18,28)(19,25)(20,26)$
2C	$2^{16}$	$4$	$2$	$16$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,11)(10,12)(13,19)(14,20)(15,18)(16,17)(21,26)(22,25)(23,27)(24,28)(29,32)(30,31)$
2D	$2^{16}$	$4$	$2$	$16$	$( 1,18)( 2,17)( 3,20)( 4,19)( 5,24)( 6,23)( 7,22)( 8,21)( 9,28)(10,27)(11,26)(12,25)(13,32)(14,31)(15,30)(16,29)$
4A1	$4^{8}$	$1$	$4$	$24$	$( 1, 8, 4, 6)( 2, 7, 3, 5)( 9,29,12,31)(10,30,11,32)(13,27,15,26)(14,28,16,25)(17,22,20,24)(18,21,19,23)$
4A-1	$4^{8}$	$1$	$4$	$24$	$( 1, 6, 4, 8)( 2, 5, 3, 7)( 9,31,12,29)(10,32,11,30)(13,26,15,27)(14,25,16,28)(17,24,20,22)(18,23,19,21)$
4B	$4^{8}$	$2$	$4$	$24$	$( 1,31, 4,29)( 2,32, 3,30)( 5,11, 7,10)( 6,12, 8, 9)(13,17,15,20)(14,18,16,19)(21,25,23,28)(22,26,24,27)$
4C	$4^{8}$	$4$	$4$	$24$	$( 1,10, 4,11)( 2, 9, 3,12)( 5,31, 7,29)( 6,32, 8,30)(13,28,15,25)(14,27,16,26)(17,23,20,21)(18,24,19,22)$
4D	$4^{8}$	$4$	$4$	$24$	$( 1,25, 4,28)( 2,26, 3,27)( 5,15, 7,13)( 6,16, 8,14)( 9,19,12,18)(10,20,11,17)(21,29,23,31)(22,30,24,32)$
8A1	$8^{4}$	$2$	$8$	$28$	$( 1,22,31,26, 4,24,29,27)( 2,21,32,25, 3,23,30,28)( 5,18,11,16, 7,19,10,14)( 6,17,12,15, 8,20, 9,13)$
8A-1	$8^{4}$	$2$	$8$	$28$	$( 1,27,29,24, 4,26,31,22)( 2,28,30,23, 3,25,32,21)( 5,14,10,19, 7,16,11,18)( 6,13, 9,20, 8,15,12,17)$
8B1	$8^{4}$	$2$	$8$	$28$	$( 1,15,31,20, 4,13,29,17)( 2,16,32,19, 3,14,30,18)( 5,28,11,21, 7,25,10,23)( 6,27,12,22, 8,26, 9,24)$
8B3	$8^{4}$	$2$	$8$	$28$	$( 1,20,29,15, 4,17,31,13)( 2,19,30,16, 3,18,32,14)( 5,21,10,28, 7,23,11,25)( 6,22, 9,27, 8,24,12,26)$

Malle's constant $a(G)$:     $1/16$

Character table

		1A	2A	2B	2C	2D	4A1	4A-1	4B	4C	4D	8A1	8A-1	8B1	8B3
	Size	1	1	2	4	4	1	1	2	4	4	2	2	2	2
	2 P	1A	1A	1A	1A	1A	2A	2A	2A	2A	2A	4B	4B	4B	4B
	Type
32.42.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
32.42.1b	R	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$1$
32.42.1c	R	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$1$	$1$	$-1$	$1$	$-1$	$1$	$-1$
32.42.1d	R	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$	$1$	$-1$	$1$	$-1$
32.42.1e	R	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$	$-1$	$1$	$-1$	$1$
32.42.1f	R	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$	$1$
32.42.1g	R	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$	$-1$	$-1$
32.42.1h	R	$1$	$1$	$1$	$1$	$-1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-1$	$-1$
32.42.2a	R	$2$	$2$	$-2$	$0$	$0$	$2$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$
32.42.2b	R	$2$	$2$	$2$	$0$	$0$	$-2$	$-2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$
32.42.2c1	C	$2$	$-2$	$0$	$0$	$0$	$-2{\zeta }_8^2$	$2{\zeta }_8^2$	$0$	$0$	$0$	$-{\zeta }_8-{\zeta }_8^3$	$-{\zeta }_8^{-1}-{\zeta }_8$	${\zeta }_8+{\zeta }_8^3$	${\zeta }_8^{-1}+{\zeta }_8$
32.42.2c2	C	$2$	$-2$	$0$	$0$	$0$	$2{\zeta }_8^2$	$-2{\zeta }_8^2$	$0$	$0$	$0$	${\zeta }_8+{\zeta }_8^3$	$-{\zeta }_8^{-1}-{\zeta }_8$	$-{\zeta }_8-{\zeta }_8^3$	${\zeta }_8^{-1}+{\zeta }_8$
32.42.2c3	C	$2$	$-2$	$0$	$0$	$0$	$-2{\zeta }_8^2$	$2{\zeta }_8^2$	$0$	$0$	$0$	${\zeta }_8+{\zeta }_8^3$	${\zeta }_8^{-1}+{\zeta }_8$	$-{\zeta }_8-{\zeta }_8^3$	$-{\zeta }_8^{-1}-{\zeta }_8$
32.42.2c4	C	$2$	$-2$	$0$	$0$	$0$	$2{\zeta }_8^2$	$-2{\zeta }_8^2$	$0$	$0$	$0$	$-{\zeta }_8-{\zeta }_8^3$	${\zeta }_8^{-1}+{\zeta }_8$	${\zeta }_8+{\zeta }_8^3$	$-{\zeta }_8^{-1}-{\zeta }_8$

Regular extensions

Data not computed