LMFDB - Galois group: 22T19

Show commands: Magma

magma: G := TransitiveGroup(22, 19);

Group action invariants

Degree $n$:	$22$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$19$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$D_{11}^2:D_5$
Parity:	$-1$	magma: IsEven(G);
Primitive:	no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:	$1$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:	(1,2,6,11,9)(3,10,5,7,4)(13,15,21,17,16)(14,18,19,22,20), (1,14,5,20)(2,21,4,13)(3,17)(6,16,11,18)(7,12,10,22)(8,19,9,15), (1,5)(2,4)(6,11)(7,10)(8,9)(12,15,18,21,13,16,19,22,14,17,20)	magma: Generators(G);

Low degree resolvents

$\card{(G/N)}$ Galois groups for stem field(s)
$2$: $C_2$ x 3
$4$: $C_2^2$
$8$: $D_{4}$
$10$: $D_{5}$
$20$: $D_{10}$
$40$: 20T7

Resolvents shown for degrees $\leq 47$

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$8$:	$D_{4}$
$10$:	$D_{5}$
$20$:	$D_{10}$
$40$:	20T7

Subfields

Degree 2: $C_2$
Degree 11: None

Low degree siblings

44T81, 44T82, 44T83
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	$1^{22}$	$1$	$1$	$0$	$()$
2A	$2^{5},1^{12}$	$22$	$2$	$5$	$(13,22)(14,21)(15,20)(16,19)(17,18)$
2B	$2^{11}$	$110$	$2$	$11$	$( 1,21)( 2,14)( 3,18)( 4,22)( 5,15)( 6,19)( 7,12)( 8,16)( 9,20)(10,13)(11,17)$
2C	$2^{10},1^{2}$	$121$	$2$	$10$	$( 1, 5)( 2, 4)( 6,11)( 7,10)( 8, 9)(12,22)(13,21)(14,20)(15,19)(16,18)$
4A	$4^{5},2$	$1210$	$4$	$16$	$( 1,14, 5,20)( 2,21, 4,13)( 3,17)( 6,16,11,18)( 7,12,10,22)( 8,19, 9,15)$
5A1	$5^{4},1^{2}$	$242$	$5$	$16$	$( 1, 5,10, 8,11)( 2, 9, 4, 6, 3)(12,14,20,16,15)(13,17,18,21,19)$
5A2	$5^{4},1^{2}$	$242$	$5$	$16$	$( 1,10,11, 5, 8)( 2, 4, 3, 9, 6)(12,20,15,14,16)(13,18,19,17,21)$
10A1	$10,5^{2},1^{2}$	$242$	$10$	$17$	$( 1,10,11, 5, 8)( 2, 4, 3, 9, 6)(12,15,21,22,13,17,14,19,18,16)$
10A3	$10,5^{2},1^{2}$	$242$	$10$	$17$	$( 1, 8, 5,11,10)( 2, 6, 9, 3, 4)(12,19,17,16,21,18,22,13,14,20)$
10B1	$10,5^{2},1^{2}$	$242$	$10$	$17$	$( 1,11, 8,10, 5)( 2, 3, 6, 4, 9)(12,20,21,17,22,13,16,15,19,14)$
10B-1	$10,5^{2},1^{2}$	$242$	$10$	$17$	$( 1, 5,10, 8,11)( 2, 9, 4, 6, 3)(12,21,16,20,19,22,13,18,14,15)$
10B3	$10^{2},1^{2}$	$242$	$10$	$18$	$( 2, 8, 6, 3, 4,11, 5, 7,10, 9)(12,20,18,13,17,16,19,21,15,22)$
10B-3	$10^{2},1^{2}$	$242$	$10$	$18$	$( 1, 7,10, 6, 4, 3, 8, 5, 9,11)(12,14,18,15,20,19,17,13,16,22)$
11A	$11^{2}$	$20$	$11$	$20$	$( 1, 8, 4,11, 7, 3,10, 6, 2, 9, 5)(12,22,21,20,19,18,17,16,15,14,13)$
11B1	$11,1^{11}$	$20$	$11$	$10$	$(12,19,15,22,18,14,21,17,13,20,16)$
11B2	$11^{2}$	$20$	$11$	$20$	$( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)(12,17,22,16,21,15,20,14,19,13,18)$
11B3	$11^{2}$	$20$	$11$	$20$	$( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,15,18,21,13,16,19,22,14,17,20)$
11B4	$11^{2}$	$20$	$11$	$20$	$( 1, 6,11, 5,10, 4, 9, 3, 8, 2, 7)(12,16,20,13,17,21,14,18,22,15,19)$
11B5	$11^{2}$	$20$	$11$	$20$	$( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)(12,21,19,17,15,13,22,20,18,16,14)$
22A	$22$	$220$	$22$	$21$	$( 1,21, 3,18, 5,15, 7,12, 9,20,11,17, 2,14, 4,22, 6,19, 8,16,10,13)$
22B1	$22$	$220$	$22$	$21$	$( 1,21, 8,16, 4,22,11,17, 7,12, 3,18,10,13, 6,19, 2,14, 9,20, 5,15)$
22B3	$22$	$220$	$22$	$21$	$( 1,21, 4,22, 7,12,10,13, 2,14, 5,15, 8,16,11,17, 3,18, 6,19, 9,20)$
22B5	$22$	$220$	$22$	$21$	$( 1,21, 7,12, 2,14, 8,16, 3,18, 9,20, 4,22,10,13, 5,15,11,17, 6,19)$
22B7	$22$	$220$	$22$	$21$	$( 1,21, 2,14, 3,18, 4,22, 5,15, 6,19, 7,12, 8,16, 9,20,10,13,11,17)$
22B9	$11,2^{5},1$	$220$	$22$	$15$	$( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)(12,18)(13,17)(14,16)(19,22)(20,21)$

Malle's constant $a(G)$: $1/5$

magma: ConjugacyClasses(G);

Group invariants

Order:	$4840=2^{3} \cdot 5 \cdot 11^{2}$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	not nilpotent
Label:	4840.bs	magma: IdentifyGroup(G);
Character table: