LMFDB - Galois group: 44T39

Show commands: Magma

magma: G := TransitiveGroup(44, 39);

Group action invariants

Degree $n$:	$44$	magma: t, n := TransitiveGroupIdentification(G); n;
Transitive number $t$:	$39$	magma: t, n := TransitiveGroupIdentification(G); t;
Group:	$D_{11}\wr C_2$
Parity:	$-1$	magma: IsEven(G);
Primitive:	no	magma: IsPrimitive(G);
magma: NilpotencyClass(G);
$\card{\Aut(F/K)}$:	$2$	magma: Order(Centralizer(SymmetricGroup(n), G));
Generators:	(1,29,17,44,11,35,6,27,22,42,15,34,9,26,4,40,20,31,13,23,7,37)(2,30,18,43,12,36,5,28,21,41,16,33,10,25,3,39,19,32,14,24,8,38), (1,20)(2,19)(3,18)(4,17)(5,16)(6,15)(7,13)(8,14)(9,11)(10,12)(23,33,44,32,42,30,40,28,37,25,35,24,34,43,31,41,29,39,27,38,26,36)	magma: Generators(G);

Low degree resolvents

|G/N| Galois groups for stem field(s)
$2$: $C_2$ x 3
$4$: $C_2^2$
$8$: $D_{4}$

Resolvents shown for degrees $\leq 47$

\|G/N\|	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$8$:	$D_{4}$

Subfields

Degree 2: $C_2$
Degree 4: $D_{4}$
Degree 11: None
Degree 22: 22T10

Low degree siblings

22T10 x 2, 44T39, 44T40 x 2, 44T41 x 2, 44T44 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, 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, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$1$	$1$	$()$
	$ 11, 11, 11, 11 $	$4$	$11$	$( 1,17,11, 6,22,15, 9, 4,20,13, 7)( 2,18,12, 5,21,16,10, 3,19,14, 8) (23,37,29,44,35,27,42,34,26,40,31)(24,38,30,43,36,28,41,33,25,39,32)$
	$ 11, 11, 11, 11 $	$4$	$11$	$( 1,11,22, 9,20, 7,17, 6,15, 4,13)( 2,12,21,10,19, 8,18, 5,16, 3,14) (23,29,35,42,26,31,37,44,27,34,40)(24,30,36,41,25,32,38,43,28,33,39)$
	$ 11, 11, 11, 11 $	$4$	$11$	$( 1,22,20,17,15,13,11, 9, 7, 6, 4)( 2,21,19,18,16,14,12,10, 8, 5, 3) (23,35,26,37,27,40,29,42,31,44,34)(24,36,25,38,28,39,30,41,32,43,33)$
	$ 11, 11, 11, 11 $	$4$	$11$	$( 1,20,15,11, 7, 4,22,17,13, 9, 6)( 2,19,16,12, 8, 3,21,18,14,10, 5) (23,26,27,29,31,34,35,37,40,42,44)(24,25,28,30,32,33,36,38,39,41,43)$
	$ 11, 11, 11, 11 $	$4$	$11$	$( 1,15, 7,22,13, 6,20,11, 4,17, 9)( 2,16, 8,21,14, 5,19,12, 3,18,10) (23,27,31,35,40,44,26,29,34,37,42)(24,28,32,36,39,43,25,30,33,38,41)$
	$ 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$4$	$11$	$(23,29,35,42,26,31,37,44,27,34,40)(24,30,36,41,25,32,38,43,28,33,39)$
	$ 11, 11, 11, 11 $	$8$	$11$	$( 1,17,11, 6,22,15, 9, 4,20,13, 7)( 2,18,12, 5,21,16,10, 3,19,14, 8) (23,44,42,40,37,35,34,31,29,27,26)(24,43,41,39,38,36,33,32,30,28,25)$
	$ 11, 11, 11, 11 $	$8$	$11$	$( 1,11,22, 9,20, 7,17, 6,15, 4,13)( 2,12,21,10,19, 8,18, 5,16, 3,14) (23,35,26,37,27,40,29,42,31,44,34)(24,36,25,38,28,39,30,41,32,43,33)$
	$ 11, 11, 11, 11 $	$8$	$11$	$( 1,22,20,17,15,13,11, 9, 7, 6, 4)( 2,21,19,18,16,14,12,10, 8, 5, 3) (23,42,37,34,29,26,44,40,35,31,27)(24,41,38,33,30,25,43,39,36,32,28)$
	$ 11, 11, 11, 11 $	$8$	$11$	$( 1, 6, 9,13,17,22, 4, 7,11,15,20)( 2, 5,10,14,18,21, 3, 8,12,16,19) (23,27,31,35,40,44,26,29,34,37,42)(24,28,32,36,39,43,25,30,33,38,41)$
	$ 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$4$	$11$	$(23,35,26,37,27,40,29,42,31,44,34)(24,36,25,38,28,39,30,41,32,43,33)$
	$ 11, 11, 11, 11 $	$8$	$11$	$( 1,17,11, 6,22,15, 9, 4,20,13, 7)( 2,18,12, 5,21,16,10, 3,19,14, 8) (23,27,31,35,40,44,26,29,34,37,42)(24,28,32,36,39,43,25,30,33,38,41)$
	$ 11, 11, 11, 11 $	$8$	$11$	$( 1,11,22, 9,20, 7,17, 6,15, 4,13)( 2,12,21,10,19, 8,18, 5,16, 3,14) (23,42,37,34,29,26,44,40,35,31,27)(24,41,38,33,30,25,43,39,36,32,28)$
	$ 11, 11, 11, 11 $	$8$	$11$	$( 1,22,20,17,15,13,11, 9, 7, 6, 4)( 2,21,19,18,16,14,12,10, 8, 5, 3) (23,26,27,29,31,34,35,37,40,42,44)(24,25,28,30,32,33,36,38,39,41,43)$
	$ 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$4$	$11$	$(23,26,27,29,31,34,35,37,40,42,44)(24,25,28,30,32,33,36,38,39,41,43)$
	$ 11, 11, 11, 11 $	$8$	$11$	$( 1,17,11, 6,22,15, 9, 4,20,13, 7)( 2,18,12, 5,21,16,10, 3,19,14, 8) (23,40,34,27,44,37,31,26,42,35,29)(24,39,33,28,43,38,32,25,41,36,30)$
	$ 11, 11, 11, 11 $	$8$	$11$	$( 1,22,20,17,15,13,11, 9, 7, 6, 4)( 2,21,19,18,16,14,12,10, 8, 5, 3) (23,37,29,44,35,27,42,34,26,40,31)(24,38,30,43,36,28,41,33,25,39,32)$
	$ 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$4$	$11$	$(23,27,31,35,40,44,26,29,34,37,42)(24,28,32,36,39,43,25,30,33,38,41)$
	$ 11, 11, 11, 11 $	$8$	$11$	$( 1,20,15,11, 7, 4,22,17,13, 9, 6)( 2,19,16,12, 8, 3,21,18,14,10, 5) (23,29,35,42,26,31,37,44,27,34,40)(24,30,36,41,25,32,38,43,28,33,39)$
	$ 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $	$4$	$11$	$(23,31,40,26,34,42,27,35,44,29,37)(24,32,39,25,33,41,28,36,43,30,38)$
	$ 22, 22 $	$44$	$22$	$( 1,29,17,44,11,35, 6,27,22,42,15,34, 9,26, 4,40,20,31,13,23, 7,37) ( 2,30,18,43,12,36, 5,28,21,41,16,33,10,25, 3,39,19,32,14,24, 8,38)$
	$ 22, 22 $	$44$	$22$	$( 1,44, 6,42, 9,40,13,37,17,35,22,34, 4,31, 7,29,11,27,15,26,20,23) ( 2,43, 5,41,10,39,14,38,18,36,21,33, 3,32, 8,30,12,28,16,25,19,24)$
	$ 22, 22 $	$44$	$22$	$( 1,35,15,40, 7,44,22,26,13,29, 6,34,20,37,11,42, 4,23,17,27, 9,31) ( 2,36,16,39, 8,43,21,25,14,30, 5,33,19,38,12,41, 3,24,18,28,10,32)$
	$ 22, 22 $	$44$	$22$	$( 1,42,13,35, 4,29,15,23, 6,40,17,34, 7,27,20,44, 9,37,22,31,11,26) ( 2,41,14,36, 3,30,16,24, 5,39,18,33, 8,28,19,43,10,38,21,32,12,25)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$22$	$2$	$( 1,34)( 2,33)( 3,43)( 4,44)( 5,32)( 6,31)( 7,42)( 8,41)( 9,29)(10,30)(11,40) (12,39)(13,27)(14,28)(15,37)(16,38)(17,26)(18,25)(19,36)(20,35)(21,24)(22,23)$
	$ 22, 22 $	$44$	$22$	$( 1,40,22,29,20,42,17,31,15,44,13,34,11,23, 9,35, 7,26, 6,37, 4,27) ( 2,39,21,30,19,41,18,32,16,43,14,33,12,24,10,36, 8,25, 5,38, 3,28)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $	$121$	$2$	$( 1,16)( 2,15)( 3,13)( 4,14)( 5,11)( 6,12)( 7,10)( 8, 9)(17,21)(18,22)(19,20) (23,33)(24,34)(25,31)(26,32)(27,30)(28,29)(35,43)(36,44)(37,41)(38,42)(39,40)$
	$ 22, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $	$44$	$22$	$( 3,21)( 4,22)( 5,19)( 6,20)( 7,17)( 8,18)( 9,15)(10,16)(11,13)(12,14) (23,32,40,25,34,41,27,36,44,30,37,24,31,39,26,33,42,28,35,43,29,38)$
	$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $	$22$	$2$	$( 1,17)( 2,18)( 3,16)( 4,15)( 5,14)( 6,13)( 7,11)( 8,12)(19,21)(20,22)(23,24) (25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)$
	$ 22, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $	$44$	$22$	$( 1,22)( 2,21)( 3,19)( 4,20)( 5,18)( 6,17)( 7,15)( 8,16)( 9,13)(10,14) (23,43,42,39,37,36,34,32,29,28,26,24,44,41,40,38,35,33,31,30,27,25)$
	$ 22, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $	$44$	$22$	$( 1,20)( 2,19)( 3,18)( 4,17)( 5,16)( 6,15)( 7,13)( 8,14)( 9,11)(10,12) (23,33,44,32,42,30,40,28,37,25,35,24,34,43,31,41,29,39,27,38,26,36)$
	$ 22, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $	$44$	$22$	$( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9,22)(10,21)(11,20)(12,19)(13,17)(14,18) (23,39,34,28,44,38,31,25,42,36,29,24,40,33,27,43,37,32,26,41,35,30)$
	$ 22, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $	$44$	$22$	$( 1, 4)( 2, 3)( 5,21)( 6,22)( 7,20)( 8,19)( 9,17)(10,18)(11,15)(12,16) (23,41,37,33,29,25,44,39,35,32,27,24,42,38,34,30,26,43,40,36,31,28)$
	$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $	$242$	$4$	$( 1,29, 2,30)( 3,41,22,40)( 4,42,21,39)( 5,32,20,27)( 6,31,19,28)( 7,44,18,38) ( 8,43,17,37)( 9,34,16,25)(10,33,15,26)(11,23,14,36)(12,24,13,35)$

magma: ConjugacyClasses(G);

Group invariants

Order:	$968=2^{3} \cdot 11^{2}$	magma: Order(G);
Cyclic:	no	magma: IsCyclic(G);
Abelian:	no	magma: IsAbelian(G);
Solvable:	yes	magma: IsSolvable(G);
Nilpotency class:	not nilpotent
Label:	968.36	magma: IdentifyGroup(G);
Character table:	35 x 35 character table

magma: CharacterTable(G);

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Group action invariants

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Group invariants

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	44T39
Degree	$44$
Order	$968$
Cyclic	no
Abelian	no
Solvable	yes
Primitive	no
$p$-group	no
Group:	$D_{11}\wr C_2$