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Properties

Label 44T9
Degree $44$
Order $88$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{44}$

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Show commands: Magma

magma: G := TransitiveGroup(44, 9);
 

Group action invariants

Degree $n$:  $44$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $9$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_{44}$
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:  (1,16,27,39,7,17,32,42,12,22,36,3,13,26,37,6,20,30,44,10,24,33,2,15,28,40,8,18,31,41,11,21,35,4,14,25,38,5,19,29,43,9,23,34), (1,10)(2,9)(3,12)(4,11)(5,8)(6,7)(13,42)(14,41)(15,43)(16,44)(17,37)(18,38)(19,40)(20,39)(21,35)(22,36)(23,33)(24,34)(25,31)(26,32)(27,30)(28,29)
magma: Generators(G);
 

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$8$:  $D_{4}$
$22$:  $D_{11}$
$44$:  $D_{22}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 11: $D_{11}$

Degree 22: $D_{22}$

Low degree siblings

44T9

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $88=2^{3} \cdot 11$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  88.5
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);