Galois group: 20T174

• Label: 20T174
• Degree: $20$
• Order: $960$
• Cyclic: no
• Abelian: no
• Solvable: no
• Primitive: no
• $p$-group: no
• Group: $D_4\times S_5$

Group action invariants

Degree $n$:		$20$
Transitive number $t$:		$174$
Group:		$D_4\times S_5$
Parity:		$-1$
Primitive:		no
$\card{\Aut(F/K)}$:		$2$
Generators:		(1,17,6,15,4,19)(2,18,5,16,3,20)(7,14)(8,13)(9,11)(10,12), (1,18,7,13,6,16,4,12,9,20)(2,17,8,14,5,15,3,11,10,19), (1,6,7)(2,5,8)(11,16,19,12,15,20)(13,14)(17,18)

Low degree resolvents

|G/N|	Galois groups for stem field(s)
$2$:	$C_2$ x 7
$4$:	$C_2^2$ x 7
$8$:	$D_{4}$ x 2, $C_2^3$
$16$:	$D_4\times C_2$
$120$:	$S_5$
$240$:	$S_5\times C_2$ x 3
$480$:	20T117

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 5: $S_5$

Degree 10: $S_5\times C_2$

Low degree siblings

20T174 x 3, 24T2665 x 4, 40T896 x 4, 40T897 x 4, 40T905 x 2, 40T917 x 2, 40T918 x 2

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	Representative
	$1^{20}$	$1$	$1$	$()$
	$2^{5},1^{10}$	$2$	$2$	$(11,12)(13,14)(15,16)(17,18)(19,20)$
	$2^{10}$	$1$	$2$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
	$2^{10}$	$2$	$2$	$( 1,16)( 2,15)( 3,17)( 4,18)( 5,19)( 6,20)( 7,12)( 8,11)( 9,13)(10,14)$
	$4^{5}$	$2$	$4$	$( 1,15, 2,16)( 3,18, 4,17)( 5,20, 6,19)( 7,11, 8,12)( 9,14,10,13)$
	$4^{5}$	$20$	$4$	$( 1,17, 2,18)( 3,16, 4,15)( 5,20, 6,19)( 7,11, 8,12)( 9,14,10,13)$
	$2^{10}$	$20$	$2$	$( 1,18)( 2,17)( 3,15)( 4,16)( 5,19)( 6,20)( 7,12)( 8,11)( 9,13)(10,14)$
	$2^{7},1^{6}$	$20$	$2$	$( 1, 3)( 2, 4)( 5, 6)( 7, 8)( 9,10)(15,17)(16,18)$
	$2^{10}$	$10$	$2$	$( 1, 3)( 2, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,18)(16,17)(19,20)$
	$2^{4},1^{12}$	$10$	$2$	$( 1, 4)( 2, 3)(15,17)(16,18)$
	$2^{10}$	$15$	$2$	$( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,10)(11,20)(12,19)(13,14)(15,18)(16,17)$
	$2^{9},1^{2}$	$30$	$2$	$( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9,10)(11,19)(12,20)(15,17)(16,18)$
	$2^{8},1^{4}$	$15$	$2$	$( 1, 4)( 2, 3)( 5, 8)( 6, 7)(11,19)(12,20)(15,17)(16,18)$
	$2^{10}$	$30$	$2$	$( 1,17)( 2,18)( 3,16)( 4,15)( 5,12)( 6,11)( 7,19)( 8,20)( 9,14)(10,13)$
	$4^{5}$	$30$	$4$	$( 1,18, 2,17)( 3,15, 4,16)( 5,11, 6,12)( 7,20, 8,19)( 9,13,10,14)$
	$3^{4},1^{8}$	$20$	$3$	$( 1, 4, 6)( 2, 3, 5)(15,17,19)(16,18,20)$
	$6,3^{2},2^{2},1^{4}$	$40$	$6$	$( 1, 4, 6)( 2, 3, 5)(11,12)(13,14)(15,18,19,16,17,20)$
	$6^{2},2^{4}$	$20$	$6$	$( 1, 3, 6, 2, 4, 5)( 7, 8)( 9,10)(11,12)(13,14)(15,18,19,16,17,20)$
	$6^{2},2^{4}$	$40$	$6$	$( 1,18, 6,16, 4,20)( 2,17, 5,15, 3,19)( 7,12)( 8,11)( 9,13)(10,14)$
	$12,4^{2}$	$40$	$12$	$( 1,17, 5,16, 4,19, 2,18, 6,15, 3,20)( 7,11, 8,12)( 9,14,10,13)$
	$3^{4},2^{4}$	$20$	$6$	$( 1, 4, 6)( 2, 3, 5)( 7, 9)( 8,10)(11,14)(12,13)(15,17,19)(16,18,20)$
	$6,3^{2},2^{4}$	$40$	$6$	$( 1, 4, 6)( 2, 3, 5)( 7, 9)( 8,10)(11,13)(12,14)(15,18,19,16,17,20)$
	$6^{2},2^{4}$	$20$	$6$	$( 1, 3, 6, 2, 4, 5)( 7,10)( 8, 9)(11,13)(12,14)(15,18,19,16,17,20)$
	$6^{2},2^{4}$	$40$	$6$	$( 1,18, 6,16, 4,20)( 2,17, 5,15, 3,19)( 7,13)( 8,14)( 9,12)(10,11)$
	$12,4^{2}$	$40$	$12$	$( 1,17, 5,16, 4,19, 2,18, 6,15, 3,20)( 7,14, 8,13)( 9,11,10,12)$
	$4^{4},2^{2}$	$30$	$4$	$( 1, 3, 6, 8)( 2, 4, 5, 7)( 9,10)(11,16,17,20)(12,15,18,19)(13,14)$
	$4^{4},2,1^{2}$	$60$	$4$	$( 1, 3, 6, 8)( 2, 4, 5, 7)( 9,10)(11,15,17,19)(12,16,18,20)$
	$4^{4},1^{4}$	$30$	$4$	$( 1, 4, 6, 7)( 2, 3, 5, 8)(11,15,17,19)(12,16,18,20)$
	$4^{4},2^{2}$	$60$	$4$	$( 1,17, 6,11)( 2,18, 5,12)( 3,20, 8,16)( 4,19, 7,15)( 9,14)(10,13)$
	$4^{5}$	$60$	$4$	$( 1,18, 5,11)( 2,17, 6,12)( 3,19, 7,16)( 4,20, 8,15)( 9,13,10,14)$
	$10^{2}$	$24$	$10$	$( 1, 3, 6, 8, 9, 2, 4, 5, 7,10)(11,13,15,18,19,12,14,16,17,20)$
	$10,5^{2}$	$48$	$10$	$( 1, 3, 6, 8, 9, 2, 4, 5, 7,10)(11,14,15,17,19)(12,13,16,18,20)$
	$5^{4}$	$24$	$5$	$( 1, 4, 6, 7, 9)( 2, 3, 5, 8,10)(11,14,15,17,19)(12,13,16,18,20)$
	$10^{2}$	$48$	$10$	$( 1,17, 6,11, 9,15, 4,19, 7,14)( 2,18, 5,12,10,16, 3,20, 8,13)$
	$20$	$48$	$20$	$( 1,18, 5,11, 9,16, 3,19, 7,13, 2,17, 6,12,10,15, 4,20, 8,14)$

Group invariants

Order:		$960=2^{6} \cdot 3 \cdot 5$
Cyclic:		no
Abelian:		no
Solvable:		no
Nilpotency class:		not nilpotent
Label:		960.10871
Character table:		35 x 35 character table