Properties

Label 44T41
Order \(968\)
n \(44\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
8:  $D_{4}$

Resolvents shown for degrees $\leq 29$

Subfields

Degree 2: $C_2$

Degree 4: $D_{4}$

Degree 11: None

Degree 22: 22T10

Low degree siblings

22T10 x 2

Siblings are shown 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 $ $1$ $1$ $()$
$ 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $11$ $( 1,17,11, 6,22,15, 9, 4,20,13, 7)( 2,18,12, 5,21,16,10, 3,19,14, 8)$
$ 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $11$ $( 1,11,22, 9,20, 7,17, 6,15, 4,13)( 2,12,21,10,19, 8,18, 5,16, 3,14)$
$ 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $11$ $( 1,22,20,17,15,13,11, 9, 7, 6, 4)( 2,21,19,18,16,14,12,10, 8, 5, 3)$
$ 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $11$ $( 1,20,15,11, 7, 4,22,17,13, 9, 6)( 2,19,16,12, 8, 3,21,18,14,10, 5)$
$ 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $4$ $11$ $( 1,15, 7,22,13, 6,20,11, 4,17, 9)( 2,16, 8,21,14, 5,19,12, 3,18,10)$
$ 11, 11, 11, 11 $ $4$ $11$ $( 1,17,11, 6,22,15, 9, 4,20,13, 7)( 2,18,12, 5,21,16,10, 3,19,14, 8) (23,37,29,44,35,27,42,34,26,40,31)(24,38,30,43,36,28,41,33,25,39,32)$
$ 11, 11, 11, 11 $ $8$ $11$ $( 1,11,22, 9,20, 7,17, 6,15, 4,13)( 2,12,21,10,19, 8,18, 5,16, 3,14) (23,37,29,44,35,27,42,34,26,40,31)(24,38,30,43,36,28,41,33,25,39,32)$
$ 11, 11, 11, 11 $ $8$ $11$ $( 1,22,20,17,15,13,11, 9, 7, 6, 4)( 2,21,19,18,16,14,12,10, 8, 5, 3) (23,37,29,44,35,27,42,34,26,40,31)(24,38,30,43,36,28,41,33,25,39,32)$
$ 11, 11, 11, 11 $ $8$ $11$ $( 1,20,15,11, 7, 4,22,17,13, 9, 6)( 2,19,16,12, 8, 3,21,18,14,10, 5) (23,37,29,44,35,27,42,34,26,40,31)(24,38,30,43,36,28,41,33,25,39,32)$
$ 11, 11, 11, 11 $ $8$ $11$ $( 1,15, 7,22,13, 6,20,11, 4,17, 9)( 2,16, 8,21,14, 5,19,12, 3,18,10) (23,37,29,44,35,27,42,34,26,40,31)(24,38,30,43,36,28,41,33,25,39,32)$
$ 11, 11, 11, 11 $ $4$ $11$ $( 1,11,22, 9,20, 7,17, 6,15, 4,13)( 2,12,21,10,19, 8,18, 5,16, 3,14) (23,29,35,42,26,31,37,44,27,34,40)(24,30,36,41,25,32,38,43,28,33,39)$
$ 11, 11, 11, 11 $ $8$ $11$ $( 1,22,20,17,15,13,11, 9, 7, 6, 4)( 2,21,19,18,16,14,12,10, 8, 5, 3) (23,29,35,42,26,31,37,44,27,34,40)(24,30,36,41,25,32,38,43,28,33,39)$
$ 11, 11, 11, 11 $ $8$ $11$ $( 1,20,15,11, 7, 4,22,17,13, 9, 6)( 2,19,16,12, 8, 3,21,18,14,10, 5) (23,29,35,42,26,31,37,44,27,34,40)(24,30,36,41,25,32,38,43,28,33,39)$
$ 11, 11, 11, 11 $ $8$ $11$ $( 1,15, 7,22,13, 6,20,11, 4,17, 9)( 2,16, 8,21,14, 5,19,12, 3,18,10) (23,29,35,42,26,31,37,44,27,34,40)(24,30,36,41,25,32,38,43,28,33,39)$
$ 11, 11, 11, 11 $ $4$ $11$ $( 1,22,20,17,15,13,11, 9, 7, 6, 4)( 2,21,19,18,16,14,12,10, 8, 5, 3) (23,35,26,37,27,40,29,42,31,44,34)(24,36,25,38,28,39,30,41,32,43,33)$
$ 11, 11, 11, 11 $ $8$ $11$ $( 1,20,15,11, 7, 4,22,17,13, 9, 6)( 2,19,16,12, 8, 3,21,18,14,10, 5) (23,35,26,37,27,40,29,42,31,44,34)(24,36,25,38,28,39,30,41,32,43,33)$
$ 11, 11, 11, 11 $ $8$ $11$ $( 1,15, 7,22,13, 6,20,11, 4,17, 9)( 2,16, 8,21,14, 5,19,12, 3,18,10) (23,35,26,37,27,40,29,42,31,44,34)(24,36,25,38,28,39,30,41,32,43,33)$
$ 11, 11, 11, 11 $ $4$ $11$ $( 1,20,15,11, 7, 4,22,17,13, 9, 6)( 2,19,16,12, 8, 3,21,18,14,10, 5) (23,26,27,29,31,34,35,37,40,42,44)(24,25,28,30,32,33,36,38,39,41,43)$
$ 11, 11, 11, 11 $ $8$ $11$ $( 1,15, 7,22,13, 6,20,11, 4,17, 9)( 2,16, 8,21,14, 5,19,12, 3,18,10) (23,26,27,29,31,34,35,37,40,42,44)(24,25,28,30,32,33,36,38,39,41,43)$
$ 11, 11, 11, 11 $ $4$ $11$ $( 1,15, 7,22,13, 6,20,11, 4,17, 9)( 2,16, 8,21,14, 5,19,12, 3,18,10) (23,27,31,35,40,44,26,29,34,37,42)(24,28,32,36,39,43,25,30,33,38,41)$
$ 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 $ $22$ $2$ $(23,28)(24,27)(25,26)(29,43)(30,44)(31,41)(32,42)(33,40)(34,39)(35,38)(36,37)$
$ 11, 11, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $44$ $22$ $( 1,17,11, 6,22,15, 9, 4,20,13, 7)( 2,18,12, 5,21,16,10, 3,19,14, 8)(23,28) (24,27)(25,26)(29,43)(30,44)(31,41)(32,42)(33,40)(34,39)(35,38)(36,37)$
$ 11, 11, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $44$ $22$ $( 1,11,22, 9,20, 7,17, 6,15, 4,13)( 2,12,21,10,19, 8,18, 5,16, 3,14)(23,28) (24,27)(25,26)(29,43)(30,44)(31,41)(32,42)(33,40)(34,39)(35,38)(36,37)$
$ 11, 11, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $44$ $22$ $( 1,22,20,17,15,13,11, 9, 7, 6, 4)( 2,21,19,18,16,14,12,10, 8, 5, 3)(23,28) (24,27)(25,26)(29,43)(30,44)(31,41)(32,42)(33,40)(34,39)(35,38)(36,37)$
$ 11, 11, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $44$ $22$ $( 1,20,15,11, 7, 4,22,17,13, 9, 6)( 2,19,16,12, 8, 3,21,18,14,10, 5)(23,28) (24,27)(25,26)(29,43)(30,44)(31,41)(32,42)(33,40)(34,39)(35,38)(36,37)$
$ 11, 11, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $44$ $22$ $( 1,15, 7,22,13, 6,20,11, 4,17, 9)( 2,16, 8,21,14, 5,19,12, 3,18,10)(23,28) (24,27)(25,26)(29,43)(30,44)(31,41)(32,42)(33,40)(34,39)(35,38)(36,37)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $121$ $2$ $( 1, 3)( 2, 4)( 5,22)( 6,21)( 7,19)( 8,20)( 9,18)(10,17)(11,16)(12,15)(13,14) (23,32)(24,31)(25,29)(26,30)(27,28)(33,44)(34,43)(35,41)(36,42)(37,39)(38,40)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $242$ $4$ $( 1,26,14,41)( 2,25,13,42)( 3,36,11,31)( 4,35,12,32)( 5,24, 9,44)( 6,23,10,43) ( 7,34, 8,33)(15,29,21,38)(16,30,22,37)(17,40,19,28)(18,39,20,27)$
$ 22, 22 $ $44$ $22$ $( 1,25,13,32, 4,38,15,43, 6,28,17,33, 7,39,20,24, 9,30,22,36,11,41) ( 2,26,14,31, 3,37,16,44, 5,27,18,34, 8,40,19,23,10,29,21,35,12,42)$
$ 22, 22 $ $44$ $22$ $( 1,25, 7,39,13,32,20,24, 4,38, 9,30,15,43,22,36, 6,28,11,41,17,33) ( 2,26, 8,40,14,31,19,23, 3,37,10,29,16,44,21,35, 5,27,12,42,18,34)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $22$ $2$ $( 1,25)( 2,26)( 3,37)( 4,38)( 5,27)( 6,28)( 7,39)( 8,40)( 9,30)(10,29)(11,41) (12,42)(13,32)(14,31)(15,43)(16,44)(17,33)(18,34)(19,23)(20,24)(21,35)(22,36)$
$ 22, 22 $ $44$ $22$ $( 1,25, 9,30,17,33, 4,38,11,41,20,24, 6,28,13,32,22,36, 7,39,15,43) ( 2,26,10,29,18,34, 3,37,12,42,19,23, 5,27,14,31,21,35, 8,40,16,44)$
$ 22, 22 $ $44$ $22$ $( 1,25, 6,28, 9,30,13,32,17,33,22,36, 4,38, 7,39,11,41,15,43,20,24) ( 2,26, 5,27,10,29,14,31,18,34,21,35, 3,37, 8,40,12,42,16,44,19,23)$
$ 22, 22 $ $44$ $22$ $( 1,25, 4,38, 6,28, 7,39, 9,30,11,41,13,32,15,43,17,33,20,24,22,36) ( 2,26, 3,37, 5,27, 8,40,10,29,12,42,14,31,16,44,18,34,19,23,21,35)$

Group invariants

Order:  $968=2^{3} \cdot 11^{2}$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [968, 36]
Character table: Data not available.