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Properties

Label 44T20
Degree $44$
Order $440$
Cyclic no
Abelian no
Solvable yes
Primitive no
$p$-group no
Group: $D_{22}:C_{10}$

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Show commands: Magma

magma: G := TransitiveGroup(44, 20);
 

Group action invariants

Degree $n$:  $44$
magma: t, n := TransitiveGroupIdentification(G); n;
 
Transitive number $t$:  $20$
magma: t, n := TransitiveGroupIdentification(G); t;
 
Group:  $D_{22}:C_{10}$
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:  (1,43,9,17,40,21,26,13,8,32,2,44,10,18,39,22,25,14,7,31)(3,41,12,19,37,23,28,15,6,29,4,42,11,20,38,24,27,16,5,30)(33,35,34,36), (1,43,16,38,29,18,23,5,25,35)(2,44,15,37,30,17,24,6,26,36)(3,42,13,39,31,19,22,7,27,34)(4,41,14,40,32,20,21,8,28,33)(9,11)(10,12)
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: 22T4

Low degree siblings

44T22

Siblings are shown with degree $\leq 47$

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

Conjugacy classes

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

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:  440.11
magma: IdentifyGroup(G);
 
Character table:

1A 2A 2B 2C 4A 5A1 5A-1 5A2 5A-2 10A1 10A-1 10A3 10A-3 10B1 10B-1 10B3 10B-3 10C1 10C-1 10C3 10C-3 11A 20A1 20A-1 20A3 20A-3 22A 22B1 22B-1
Size 1 1 2 22 22 11 11 11 11 11 11 11 11 22 22 22 22 22 22 22 22 10 22 22 22 22 10 10 10
2 P 1A 1A 1A 1A 2A 5A2 5A1 5A-2 5A-1 5A1 5A-1 5A2 5A-2 5A-1 5A-1 5A1 5A2 5A1 5A-2 5A-2 5A2 11A 10A-1 10A1 10A-3 10A3 11A 11A 11A
5 P 1A 2A 2B 2C 4A 1A 1A 1A 1A 2A 2A 2A 2A 2B 2C 2C 2C 2B 2C 2B 2B 11A 4A 4A 4A 4A 22A 22B1 22B-1
11 P 1A 2A 2B 2C 4A 5A1 5A-2 5A-1 5A2 10A-3 10A3 10A-1 10A1 10C-1 10B-1 10B1 10B-3 10C1 10B3 10C3 10C-3 1A 20A-1 20A1 20A-3 20A3 2A 2B 2B
Type
440.11.1a R 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
440.11.1b R 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
440.11.1c R 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
440.11.1d R 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
440.11.1e1 C 1 1 1 1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 1 ζ5 ζ5−1 ζ5−2 ζ52 1 1 1
440.11.1e2 C 1 1 1 1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 1 ζ5−1 ζ5 ζ52 ζ5−2 1 1 1
440.11.1e3 C 1 1 1 1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 1 ζ5−2 ζ52 ζ5−1 ζ5 1 1 1
440.11.1e4 C 1 1 1 1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 1 ζ52 ζ5−2 ζ5 ζ5−1 1 1 1
440.11.1f1 C 1 1 −1 −1 1 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 −ζ5−1 −ζ5 −ζ52 −ζ5−2 −ζ5−1 −ζ5 −ζ52 −ζ5−2 1 ζ5 ζ5−1 ζ5−2 ζ52 1 −1 −1
440.11.1f2 C 1 1 −1 −1 1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 −ζ5 −ζ5−1 −ζ5−2 −ζ52 −ζ5 −ζ5−1 −ζ5−2 −ζ52 1 ζ5−1 ζ5 ζ52 ζ5−2 1 −1 −1
440.11.1f3 C 1 1 −1 −1 1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 −ζ52 −ζ5−2 −ζ5 −ζ5−1 −ζ52 −ζ5−2 −ζ5 −ζ5−1 1 ζ5−2 ζ52 ζ5−1 ζ5 1 −1 −1
440.11.1f4 C 1 1 −1 −1 1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 −ζ5−2 −ζ52 −ζ5−1 −ζ5 −ζ5−2 −ζ52 −ζ5−1 −ζ5 1 ζ52 ζ5−2 ζ5 ζ5−1 1 −1 −1
440.11.1g1 C 1 1 −1 1 −1 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 ζ5−1 ζ5 ζ52 ζ5−2 −ζ5−1 −ζ5 −ζ52 −ζ5−2 1 −ζ5 −ζ5−1 −ζ5−2 −ζ52 1 −1 −1
440.11.1g2 C 1 1 −1 1 −1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 ζ5 ζ5−1 ζ5−2 ζ52 −ζ5 −ζ5−1 −ζ5−2 −ζ52 1 −ζ5−1 −ζ5 −ζ52 −ζ5−2 1 −1 −1
440.11.1g3 C 1 1 −1 1 −1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 ζ52 ζ5−2 ζ5 ζ5−1 −ζ52 −ζ5−2 −ζ5 −ζ5−1 1 −ζ5−2 −ζ52 −ζ5−1 −ζ5 1 −1 −1
440.11.1g4 C 1 1 −1 1 −1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 ζ5−2 ζ52 ζ5−1 ζ5 −ζ5−2 −ζ52 −ζ5−1 −ζ5 1 −ζ52 −ζ5−2 −ζ5 −ζ5−1 1 −1 −1
440.11.1h1 C 1 1 1 −1 −1 ζ5−2 ζ52 ζ5 ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 −ζ5−1 −ζ5 −ζ52 −ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 1 −ζ5 −ζ5−1 −ζ5−2 −ζ52 1 1 1
440.11.1h2 C 1 1 1 −1 −1 ζ52 ζ5−2 ζ5−1 ζ5 ζ5−2 ζ52 ζ5−1 ζ5 −ζ5 −ζ5−1 −ζ5−2 −ζ52 ζ5 ζ5−1 ζ5−2 ζ52 1 −ζ5−1 −ζ5 −ζ52 −ζ5−2 1 1 1
440.11.1h3 C 1 1 1 −1 −1 ζ5−1 ζ5 ζ5−2 ζ52 ζ5 ζ5−1 ζ5−2 ζ52 −ζ52 −ζ5−2 −ζ5 −ζ5−1 ζ52 ζ5−2 ζ5 ζ5−1 1 −ζ5−2 −ζ52 −ζ5−1 −ζ5 1 1 1
440.11.1h4 C 1 1 1 −1 −1 ζ5 ζ5−1 ζ52 ζ5−2 ζ5−1 ζ5 ζ52 ζ5−2 −ζ5−2 −ζ52 −ζ5−1 −ζ5 ζ5−2 ζ52 ζ5−1 ζ5 1 −ζ52 −ζ5−2 −ζ5 −ζ5−1 1 1 1
440.11.2a R 2 −2 0 0 0 2 2 2 2 −2 −2 −2 −2 0 0 0 0 0 0 0 0 2 0 0 0 0 −2 0 0
440.11.2b1 C 2 −2 0 0 0 2ζ5−2 2ζ52 2ζ5 2ζ5−1 −2ζ52 −2ζ5−2 −2ζ5 −2ζ5−1 0 0 0 0 0 0 0 0 2 0 0 0 0 −2 0 0
440.11.2b2 C 2 −2 0 0 0 2ζ52 2ζ5−2 2ζ5−1 2ζ5 −2ζ5−2 −2ζ52 −2ζ5−1 −2ζ5 0 0 0 0 0 0 0 0 2 0 0 0 0 −2 0 0
440.11.2b3 C 2 −2 0 0 0 2ζ5−1 2ζ5 2ζ5−2 2ζ52 −2ζ5 −2ζ5−1 −2ζ5−2 −2ζ52 0 0 0 0 0 0 0 0 2 0 0 0 0 −2 0 0
440.11.2b4 C 2 −2 0 0 0 2ζ5 2ζ5−1 2ζ52 2ζ5−2 −2ζ5−1 −2ζ5 −2ζ52 −2ζ5−2 0 0 0 0 0 0 0 0 2 0 0 0 0 −2 0 0
440.11.10a R 10 10 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 −1 −1 −1
440.11.10b R 10 10 −10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 −1 1 1
440.11.10c1 C 10 −10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 1 −2ζ11−2−1−2ζ11−2ζ113−2ζ114−2ζ115 2ζ11−2+1+2ζ11+2ζ113+2ζ114+2ζ115
440.11.10c2 C 10 −10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 −1 0 0 0 0 1 2ζ11−2+1+2ζ11+2ζ113+2ζ114+2ζ115 −2ζ11−2−1−2ζ11−2ζ113−2ζ114−2ζ115

magma: CharacterTable(G);