Properties

Label 44T6
Order \(88\)
n \(44\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $D_{22}:C_2$

Learn more about

Group action invariants

Degree $n$ :  $44$
Transitive number $t$ :  $6$
Group :  $D_{22}:C_2$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,34,2,33)(3,35,4,36)(5,32,6,31)(7,30,8,29)(9,27,10,28)(11,26,12,25)(13,24,14,23)(15,21,16,22)(17,19,18,20)(37,41,38,42)(39,44,40,43), (1,39,31,24,15,7,41,36,28,19,12)(2,40,32,23,16,8,42,35,27,20,11)(3,37,30,22,14,6,44,33,25,17,9,4,38,29,21,13,5,43,34,26,18,10)
$|\Aut(F/K)|$:  $22$

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 $\leq 29$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 11: $D_{11}$

Degree 22: 22T2

Low degree siblings

There are no siblings with degree $\leq 29$
Data on whether or not a number field with this Galois group has arithmetically equivalent fields has not been computed.

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

Group invariants

Order:  $88=2^{3} \cdot 11$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [88, 7]
Character table: Data not available.