LMFDB - Galois group: 20T157

Show commands: Magma

magma:G := TransitiveGroup(20, 157);

Group invariants

Abstract group:	$D_5^2.D_4$	magma:IdentifyGroup(G);
Order:	$800=2^{5} \cdot 5^{2}$	magma:Order(G);
Cyclic:	no	magma:IsCyclic(G);
Abelian:	no	magma:IsAbelian(G);
Solvable:	yes	magma:IsSolvable(G);
Nilpotency class:	not nilpotent	magma:NilpotencyClass(G);

Group action invariants

Degree $n$:	$20$	magma:t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$157$	magma:t, n := TransitiveGroupIdentification(G); t;
Parity:	$-1$	magma:IsEven(G);
Primitive:	no	magma:IsPrimitive(G);
$\card{\Aut(F/K)}$:	$2$	magma:Order(Centralizer(SymmetricGroup(n), G));
Generators:	$(1,9)(2,10)(13,18)(14,17)$, $(3,20,16,11,7)(4,19,15,12,8)$, $(1,14,9,17)(2,13,10,18)(3,20,7,11)(4,19,8,12)(5,6)$, $(1,17,13,10,6,2,18,14,9,5)(3,4)(7,8)(11,12)(15,16)(19,20)$, $(1,12)(2,11)(3,14)(4,13)(5,16)(6,15)(7,17)(8,18)(9,19)(10,20)$	magma:Generators(G);

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, $C_2^3$
$16$: $D_4\times C_2$, $Q_8:C_2$ x 2
$32$: 16T37
$400$: $(D_5 \wr C_2):C_2$

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 7
$4$:	$C_2^2$ x 7
$8$:	$D_{4}$ x 2, $C_2^3$
$16$:	$D_4\times C_2$, $Q_8:C_2$ x 2
$32$:	16T37
$400$:	$(D_5 \wr C_2):C_2$

Subfields

Degree 2: $C_2$
Degree 4: $D_{4}$
Degree 5: None
Degree 10: $(D_5 \wr C_2):C_2$

Low degree siblings

20T157 x 3, 40T682, 40T687, 40T694 x 2, 40T725 x 2, 40T728, 40T729, 40T739 x 2, 40T746, 40T753, 40T759 x 2
Siblings are shown with degree $\leq 47$

A number field with this Galois group has exactly one arithmetically equivalent field.

Conjugacy classes

Label	Cycle Type	Size	Order	Index	Representative
1A	$1^{20}$	$1$	$1$	$0$	$()$
2A	$2^{10}$	$1$	$2$	$10$	$( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)$
2B	$2^{10}$	$10$	$2$	$10$	$( 1, 2)( 3, 8)( 4, 7)( 5, 6)( 9,10)(11,19)(12,20)(13,14)(15,16)(17,18)$
2C	$2^{4},1^{12}$	$10$	$2$	$4$	$( 1,18)( 2,17)( 5,14)( 6,13)$
2D	$2^{10}$	$20$	$2$	$10$	$( 1,19)( 2,20)( 3, 5)( 4, 6)( 7,10)( 8, 9)(11,14)(12,13)(15,18)(16,17)$
2E	$2^{8},1^{4}$	$25$	$2$	$8$	$( 1, 9)( 2,10)( 3,11)( 4,12)(13,18)(14,17)(15,19)(16,20)$
2F	$2^{10}$	$25$	$2$	$10$	$( 1, 2)( 3,12)( 4,11)( 5,18)( 6,17)( 7, 8)( 9,14)(10,13)(15,20)(16,19)$
4A	$4^{5}$	$20$	$4$	$15$	$( 1, 3, 2, 4)( 5,12, 6,11)( 7,14, 8,13)( 9,20,10,19)(15,18,16,17)$
4B1	$4^{4},2,1^{2}$	$50$	$4$	$13$	$( 1,17, 9,14)( 2,18,10,13)( 3,20,11,16)( 4,19,12,15)( 5, 6)$
4B-1	$4^{4},2,1^{2}$	$50$	$4$	$13$	$( 1,14, 9,17)( 2,13,10,18)( 3,16,11,20)( 4,15,12,19)( 5, 6)$
4C1	$4^{4},2,1^{2}$	$50$	$4$	$13$	$( 1,14,18, 5)( 2,13,17, 6)( 3,20, 7,11)( 4,19, 8,12)( 9,10)$
4C-1	$4^{4},2,1^{2}$	$50$	$4$	$13$	$( 1, 5,18,14)( 2, 6,17,13)( 3,11, 7,20)( 4,12, 8,19)( 9,10)$
4D	$4^{4},2^{2}$	$100$	$4$	$14$	$( 1,19,13,12)( 2,20,14,11)( 3,10, 7, 5)( 4, 9, 8, 6)(15,18)(16,17)$
4E	$4^{5}$	$100$	$4$	$15$	$( 1, 8, 2, 7)( 3, 9,12,14)( 4,10,11,13)( 5,20,18,15)( 6,19,17,16)$
5A	$5^{4}$	$8$	$5$	$16$	$( 1, 6, 9,13,18)( 2, 5,10,14,17)( 3, 7,11,16,20)( 4, 8,12,15,19)$
5B	$5^{2},1^{10}$	$8$	$5$	$8$	$( 1, 9,18, 6,13)( 2,10,17, 5,14)$
5C	$5^{4}$	$8$	$5$	$16$	$( 1,18,13, 9, 6)( 2,17,14,10, 5)( 3,16, 7,20,11)( 4,15, 8,19,12)$
10A	$10^{2}$	$8$	$10$	$18$	$( 1,14, 6,17, 9, 2,13, 5,18,10)( 3,15, 7,19,11, 4,16, 8,20,12)$
10B	$10,2^{5}$	$8$	$10$	$14$	$( 1, 5, 9,14,18, 2, 6,10,13,17)( 3, 4)( 7, 8)(11,12)(15,16)(19,20)$
10C	$10^{2}$	$8$	$10$	$18$	$( 1,10,18, 5,13, 2, 9,17, 6,14)( 3,19,16,12, 7, 4,20,15,11, 8)$
10D	$10,2^{5}$	$40$	$10$	$14$	$( 1,10,18, 5,13, 2, 9,17, 6,14)( 3, 8)( 4, 7)(11,19)(12,20)(15,16)$
10E	$5^{2},2^{4},1^{2}$	$40$	$10$	$12$	$( 1,18)( 2,17)( 3,20,16,11, 7)( 4,19,15,12, 8)( 5,14)( 6,13)$
10F1	$10^{2}$	$40$	$10$	$18$	$( 1, 8,18, 4,13,19, 9,15, 6,12)( 2, 7,17, 3,14,20,10,16, 5,11)$
10F3	$10^{2}$	$40$	$10$	$18$	$( 1, 4, 9,12,18,19, 6, 8,13,15)( 2, 3,10,11,17,20, 5, 7,14,16)$
20A1	$20$	$40$	$20$	$19$	$( 1,11,10, 8,18, 3, 5,19,13,16, 2,12, 9, 7,17, 4, 6,20,14,15)$
20A-1	$20$	$40$	$20$	$19$	$( 1,15,14,20, 6, 4,17, 7, 9,12, 2,16,13,19, 5, 3,18, 8,10,11)$

Malle's constant $a(G)$: $1/4$

magma:ConjugacyClasses(G);

Character table

		1A	2A	2B	2C	2D	2E	2F	4A	4B1	4B-1	4C1	4C-1	4D	4E	5A	5B	5C	10A	10B	10C	10D	10E	10F1	10F3	20A1	20A-1
	Size	1	1	10	10	20	25	25	20	50	50	50	50	100	100	8	8	8	8	8	8	40	40	40	40	40	40
	2 P	1A	1A	1A	1A	1A	1A	1A	2A	2E	2E	2E	2E	2E	2F	5A	5B	5C	5A	5B	5C	5B	5B	5A	5A	10C	10C
	5 P	1A	2A	2B	2C	2D	2E	2F	4A	4B1	4B-1	4C1	4C-1	4D	4E	1A	1A	1A	2A	2A	2A	2B	2C	2D	2D	4A	4A
	Type
800.970.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$
800.970.1b	R	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$
800.970.1c	R	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$
800.970.1d	R	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$
800.970.1e	R	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$
800.970.1f	R	$1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$
800.970.1g	R	$1$	$1$	$1$	$1$	$- 1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$1$	$1$
800.970.1h	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$	$- 1$	$- 1$	$- 1$	$1$	$- 1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$1$	$- 1$	$- 1$
800.970.2a	R	$2$	$- 2$	$- 2$	$2$	$0$	$2$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$2$	$2$	$2$	$- 2$	$- 2$	$- 2$	$- 2$	$2$	$0$	$0$	$0$	$0$
800.970.2b	R	$2$	$- 2$	$2$	$- 2$	$0$	$2$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$2$	$2$	$2$	$- 2$	$- 2$	$- 2$	$2$	$- 2$	$0$	$0$	$0$	$0$
800.970.2c1	C	$2$	$- 2$	$0$	$0$	$0$	$- 2$	$2$	$0$	$0$	$0$	$- 2 i$	$2 i$	$0$	$0$	$2$	$2$	$2$	$- 2$	$- 2$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$
800.970.2c2	C	$2$	$- 2$	$0$	$0$	$0$	$- 2$	$2$	$0$	$0$	$0$	$2 i$	$- 2 i$	$0$	$0$	$2$	$2$	$2$	$- 2$	$- 2$	$- 2$	$0$	$0$	$0$	$0$	$0$	$0$
800.970.2d1	C	$2$	$2$	$0$	$0$	$0$	$- 2$	$- 2$	$0$	$- 2 i$	$2 i$	$0$	$0$	$0$	$0$	$2$	$2$	$2$	$2$	$2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$
800.970.2d2	C	$2$	$2$	$0$	$0$	$0$	$- 2$	$- 2$	$0$	$2 i$	$- 2 i$	$0$	$0$	$0$	$0$	$2$	$2$	$2$	$2$	$2$	$2$	$0$	$0$	$0$	$0$	$0$	$0$
800.970.8a	R	$8$	$8$	$0$	$0$	$0$	$0$	$0$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$- 2$	$- 2$	$3$	$- 2$	$- 2$	$3$	$0$	$0$	$0$	$0$	$- 1$	$- 1$
800.970.8b	R	$8$	$8$	$0$	$0$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$3$	$- 2$	$- 2$	$3$	$- 2$	$- 2$	$0$	$0$	$- 1$	$- 1$	$0$	$0$
800.970.8c	R	$8$	$8$	$4$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 2$	$3$	$- 2$	$- 2$	$3$	$- 2$	$- 1$	$- 1$	$0$	$0$	$0$	$0$
800.970.8d	R	$8$	$8$	$- 4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 2$	$3$	$- 2$	$- 2$	$3$	$- 2$	$1$	$1$	$0$	$0$	$0$	$0$
800.970.8e	R	$8$	$8$	$0$	$0$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$3$	$- 2$	$- 2$	$3$	$- 2$	$- 2$	$0$	$0$	$1$	$1$	$0$	$0$
800.970.8f	R	$8$	$8$	$0$	$0$	$0$	$0$	$0$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$- 2$	$- 2$	$3$	$- 2$	$- 2$	$3$	$0$	$0$	$0$	$0$	$1$	$1$
800.970.8g	R	$8$	$- 8$	$- 4$	$4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 2$	$3$	$- 2$	$2$	$- 3$	$2$	$1$	$- 1$	$0$	$0$	$0$	$0$
800.970.8h	R	$8$	$- 8$	$4$	$- 4$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 2$	$3$	$- 2$	$2$	$- 3$	$2$	$- 1$	$1$	$0$	$0$	$0$	$0$
800.970.8i1	R	$8$	$- 8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$3$	$- 2$	$- 2$	$- 3$	$2$	$2$	$0$	$0$	$2 ζ_{5}^{- 2} + 1 + 2 ζ_{5}^{2}$	$- 2 ζ_{5}^{- 2} - 1 - 2 ζ_{5}^{2}$	$0$	$0$
800.970.8i2	R	$8$	$- 8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$3$	$- 2$	$- 2$	$- 3$	$2$	$2$	$0$	$0$	$- 2 ζ_{5}^{- 2} - 1 - 2 ζ_{5}^{2}$	$2 ζ_{5}^{- 2} + 1 + 2 ζ_{5}^{2}$	$0$	$0$
800.970.8j1	C	$8$	$- 8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 2$	$- 2$	$3$	$2$	$2$	$- 3$	$0$	$0$	$0$	$0$	$- 2 ζ_{20}^{3} + ζ_{20}^{5} - 2 ζ_{20}^{7}$	$2 ζ_{20}^{3} - ζ_{20}^{5} + 2 ζ_{20}^{7}$
800.970.8j2	C	$8$	$- 8$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$0$	$- 2$	$- 2$	$3$	$2$	$2$	$- 3$	$0$	$0$	$0$	$0$	$2 ζ_{20}^{3} - ζ_{20}^{5} + 2 ζ_{20}^{7}$	$- 2 ζ_{20}^{3} + ζ_{20}^{5} - 2 ζ_{20}^{7}$

magma:CharacterTable(G);

Regular extensions

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	20T157

Degree	$20$
Order	$800$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$D_5^2.D_4$