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Properties

Label 44T23
Degree $44$
Order $440$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $C_{44}:C_{10}$

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Show commands: Magma

magma: G := TransitiveGroup(44, 23);
 

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
$2$:  $C_2$ x 3
$4$:  $C_2^2$
$5$:  $C_5$
$8$:  $D_{4}$
$10$:  $C_{10}$ x 3
$20$:  20T3
$40$:  20T12
$110$:  $F_{11}$
$220$:  22T6

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 11: $F_{11}$

Degree 22: 22T6

Low degree siblings

44T23

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

magma: ConjugacyClasses(G);
 

Group invariants

Order:  $440=2^{3} \cdot 5 \cdot 11$
magma: Order(G);
 
Cyclic:  no
magma: IsCyclic(G);
 
Abelian:  no
magma: IsAbelian(G);
 
Solvable:  yes
magma: IsSolvable(G);
 
Nilpotency class:   not nilpotent
Label:  440.9
magma: IdentifyGroup(G);
 
Character table: not available.

magma: CharacterTable(G);