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Properties

Label 44T22
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, 22);
 

Group action invariants

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

44T20

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

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:  440.11
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);