Properties

Label 44T31
Order \(660\)
n \(44\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No

Learn more about

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
3:  $C_3$
5:  $C_5$
12:  $A_4$
15:  $C_{15}$
55:  $C_{11}:C_5$
60:  20T14

Resolvents shown for degrees $\leq 29$

Subfields

Degree 2: None

Degree 4: $A_4$

Degree 11: $C_{11}:C_5$

Degree 22: None

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

Group invariants

Order:  $660=2^{2} \cdot 3 \cdot 5 \cdot 11$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [660, 16]
Character table: Data not available.