Galois group: 32T2134

• Label: 32T2134
• Degree: $32$
• Order: $160$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• pp-group: no
• Group: $C_2.C_2^4:C_5$

Group invariants

Abstract group:		$C_2.C_2^4:C_5$
Order:		$160=2^{5} \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$32$
Transitive number $t$:		$2134$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,12,6,30,28,2,11,5,29,27)(3,31,18,7,20,4,32,17,8,19)(9,10)(13,23,16,21,25,14,24,15,22,26)$, $(1,18,13,20,5)(2,17,14,19,6)(3,26,24,31,9)(4,25,23,32,10)(7,28,16,22,30)(8,27,15,21,29)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$5$:	$C_5$
$80$:	$C_2^4 : C_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 4: None

Degree 8: None

Degree 16: 16T178

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	$2^{16}$	$1$	$2$	$16$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22)(23,24)(25,26)(27,28)(29,30)(31,32)$
2B	$2^{16}$	$10$	$2$	$16$	$( 1,22)( 2,21)( 3,16)( 4,15)( 5,17)( 6,18)( 7,28)( 8,27)( 9,29)(10,30)(11,23)(12,24)(13,26)(14,25)(19,31)(20,32)$
4A	$4^{8}$	$10$	$4$	$24$	$( 1,18, 2,17)( 3,19, 4,20)( 5,22, 6,21)( 7,24, 8,23)( 9,25,10,26)(11,28,12,27)(13,29,14,30)(15,32,16,31)$
4B	$4^{8}$	$10$	$4$	$24$	$( 1,26, 2,25)( 3,28, 4,27)( 5,29, 6,30)( 7,16, 8,15)( 9,17,10,18)(11,19,12,20)(13,22,14,21)(23,32,24,31)$
5A1	$5^{6},1^{2}$	$16$	$5$	$24$	$( 1,23, 8,26, 5)( 2,24, 7,25, 6)( 3,11,31,21,16)( 4,12,32,22,15)( 9,19,30,27,18)(10,20,29,28,17)$
5A-1	$5^{6},1^{2}$	$16$	$5$	$24$	$( 1, 5,26, 8,23)( 2, 6,25, 7,24)( 3,16,21,31,11)( 4,15,22,32,12)( 9,18,27,30,19)(10,17,28,29,20)$
5A2	$5^{6},1^{2}$	$16$	$5$	$24$	$( 1, 8, 5,23,26)( 2, 7, 6,24,25)( 3,31,16,11,21)( 4,32,15,12,22)( 9,30,18,19,27)(10,29,17,20,28)$
5A-2	$5^{6},1^{2}$	$16$	$5$	$24$	$( 1,26,23, 5, 8)( 2,25,24, 6, 7)( 3,21,11,16,31)( 4,22,12,15,32)( 9,27,19,18,30)(10,28,20,17,29)$
10A1	$10^{3},2$	$16$	$10$	$28$	$( 1,25,23, 6, 8, 2,26,24, 5, 7)( 3,22,11,15,31, 4,21,12,16,32)( 9,28,19,17,30,10,27,20,18,29)(13,14)$
10A-1	$10^{3},2$	$16$	$10$	$28$	$( 1, 7, 5,24,26, 2, 8, 6,23,25)( 3,32,16,12,21, 4,31,15,11,22)( 9,29,18,20,27,10,30,17,19,28)(13,14)$
10A3	$10^{3},2$	$16$	$10$	$28$	$( 1, 6,26, 7,23, 2, 5,25, 8,24)( 3,15,21,32,11, 4,16,22,31,12)( 9,17,27,29,19,10,18,28,30,20)(13,14)$
10A-3	$10^{3},2$	$16$	$10$	$28$	$( 1,24, 8,25, 5, 2,23, 7,26, 6)( 3,12,31,22,16, 4,11,32,21,15)( 9,20,30,28,18,10,19,29,27,17)(13,14)$

Malle's constant $a(G)$:     $1/16$

Character table

		1A	2A	2B	4A	4B	5A1	5A-1	5A2	5A-2	10A1	10A-1	10A3	10A-3
	Size	1	1	10	10	10	16	16	16	16	16	16	16	16
	2 P	1A	1A	1A	2A	2A	5A2	5A-2	5A-1	5A1	5A1	5A-1	5A-2	5A2
	5 P	1A	2A	2B	4A	4B	1A	1A	1A	1A	2A	2A	2A	2A
	Type
160.199.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
160.199.1b1	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^2$	${\zeta }_5^{-2}$
160.199.1b2	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5^{-2}$	${\zeta }_5^2$
160.199.1b3	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5$	${\zeta }_5^{-1}$
160.199.1b4	C	$1$	$1$	$1$	$1$	$1$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5^{-1}$	${\zeta }_5$
160.199.4a	S	$4$	$-4$	$0$	$0$	$0$	$-1$	$-1$	$-1$	$-1$	$1$	$1$	$1$	$1$
160.199.4b1	C	$4$	$-4$	$0$	$0$	$0$	$-{\zeta }_5$	$-{\zeta }_5^{-1}$	$-{\zeta }_5^{-2}$	$-{\zeta }_5^2$	${\zeta }_5^2$	${\zeta }_5^{-2}$	${\zeta }_5^{-1}$	${\zeta }_5$
160.199.4b2	C	$4$	$-4$	$0$	$0$	$0$	$-{\zeta }_5^{-1}$	$-{\zeta }_5$	$-{\zeta }_5^2$	$-{\zeta }_5^{-2}$	${\zeta }_5^{-2}$	${\zeta }_5^2$	${\zeta }_5$	${\zeta }_5^{-1}$
160.199.4b3	C	$4$	$-4$	$0$	$0$	$0$	$-{\zeta }_5^2$	$-{\zeta }_5^{-2}$	$-{\zeta }_5$	$-{\zeta }_5^{-1}$	${\zeta }_5^{-1}$	${\zeta }_5$	${\zeta }_5^{-2}$	${\zeta }_5^2$
160.199.4b4	C	$4$	$-4$	$0$	$0$	$0$	$-{\zeta }_5^{-2}$	$-{\zeta }_5^2$	$-{\zeta }_5^{-1}$	$-{\zeta }_5$	${\zeta }_5$	${\zeta }_5^{-1}$	${\zeta }_5^2$	${\zeta }_5^{-2}$
160.199.5a	R	$5$	$5$	$-3$	$1$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
160.199.5b	R	$5$	$5$	$1$	$-3$	$1$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$
160.199.5c	R	$5$	$5$	$1$	$1$	$-3$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

Data not computed