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Properties

Label 44T8
Degree $44$
Order $88$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{11}:D_4$

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Show commands: Magma

magma: G := TransitiveGroup(44, 8);
 

Group action invariants

Degree $n$:  $44$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $8$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $C_{11}:D_4$
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:  (1,44)(2,43)(3,41)(4,42)(5,39)(6,40)(7,38)(8,37)(9,33)(10,34)(11,36)(12,35)(13,32)(14,31)(15,29)(16,30)(17,25)(18,26)(19,27)(20,28)(23,24), (1,29,14,42,27,10,37,22,7,34,20,4,32,15,43,26,12,39,24,5,36,18)(2,30,13,41,28,9,38,21,8,33,19,3,31,16,44,25,11,40,23,6,35,17)
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

44T6

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

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:  88.7
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);