Galois group: 10T24

• Label: 10T24
• Degree: $10$
• Order: $320$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: no
• Group: $(C_2^4 : C_5):C_4$

Group invariants

Abstract group:		$(C_2^4 : C_5):C_4$
Order:		$320=2^{6} \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		not nilpotent

Group action invariants

Degree $n$:		$10$
Transitive number $t$:		$24$
CHM label:		$[2^{4}]F(5)$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		$(1,7,9,3)(2,4,8,6)$, $(1,3,5,7,9)(2,4,6,8,10)$, $(2,7)(5,10)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$
$4$:	$C_4$
$20$:	$F_5$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: None

Degree 5: $F_5$

Low degree siblings

10T25, 16T711, 20T77, 20T78, 20T79, 20T80, 20T83, 20T88, 32T9312, 40T206, 40T207, 40T296, 40T297, 40T298, 40T299, 40T300, 40T301, 40T302, 40T303

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	$1^{10}$	$1$	$1$	$0$	$()$
2A	$2^{4},1^{2}$	$5$	$2$	$4$	$( 2, 7)( 3, 8)( 4, 9)( 5,10)$
2B	$2^{2},1^{6}$	$10$	$2$	$2$	$(3,8)(4,9)$
2C	$2^{4},1^{2}$	$20$	$2$	$4$	$( 2,10)( 3, 9)( 4, 8)( 5, 7)$
4A	$4^{2},1^{2}$	$20$	$4$	$6$	$( 2, 5, 7,10)( 3, 9, 8, 4)$
4B	$4,2^{3}$	$40$	$4$	$6$	$( 1, 6)( 2, 5)( 3, 4, 8, 9)( 7,10)$
4C1	$4^{2},1^{2}$	$40$	$4$	$6$	$( 2, 9,10, 3)( 4, 5, 8, 7)$
4C-1	$4^{2},1^{2}$	$40$	$4$	$6$	$( 2, 3,10, 9)( 4, 7, 8, 5)$
5A	$5^{2}$	$64$	$5$	$8$	$( 1, 9, 2,10, 8)( 3, 6, 4, 7, 5)$
8A1	$8,2$	$40$	$8$	$8$	$( 1, 6)( 2, 8, 5, 4, 7, 3,10, 9)$
8A-1	$8,2$	$40$	$8$	$8$	$( 1, 6)( 2, 9,10, 3, 7, 4, 5, 8)$

Malle's constant $a(G)$:     $1/2$

Character table

		1A	2A	2B	2C	4A	4B	4C1	4C-1	5A	8A1	8A-1
	Size	1	5	10	20	20	40	40	40	64	40	40
	2 P	1A	1A	1A	1A	2A	2B	2C	2C	5A	4A	4A
	5 P	1A	2A	2B	2C	4A	4B	4C1	4C-1	1A	8A1	8A-1
	Type
320.1635.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
320.1635.1b	R	$1$	$1$	$1$	$1$	$1$	$1$	$-1$	$-1$	$1$	$-1$	$-1$
320.1635.1c1	C	$1$	$1$	$1$	$-1$	$-1$	$-1$	$-i$	$i$	$1$	$i$	$-i$
320.1635.1c2	C	$1$	$1$	$1$	$-1$	$-1$	$-1$	$i$	$-i$	$1$	$-i$	$i$
320.1635.4a	R	$4$	$4$	$4$	$0$	$0$	$0$	$0$	$0$	$-1$	$0$	$0$
320.1635.5a	R	$5$	$-3$	$1$	$1$	$1$	$-1$	$1$	$1$	$0$	$-1$	$-1$
320.1635.5b	R	$5$	$-3$	$1$	$1$	$1$	$-1$	$-1$	$-1$	$0$	$1$	$1$
320.1635.5c1	C	$5$	$-3$	$1$	$-1$	$-1$	$1$	$-i$	$i$	$0$	$-i$	$i$
320.1635.5c2	C	$5$	$-3$	$1$	$-1$	$-1$	$1$	$i$	$-i$	$0$	$i$	$-i$
320.1635.10a	R	$10$	$2$	$-2$	$2$	$-2$	$0$	$0$	$0$	$0$	$0$	$0$
320.1635.10b	R	$10$	$2$	$-2$	$-2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$

Regular extensions

$f_{ 1 } =$

$x^{10} - 15 x^{8} + 100 x^{6} + \left(-5 t - 250\right) x^{4} + 30 t x^{2} - t^{2}$