Properties

Label 44T47
Order \(1320\)
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 $t$ :  $47$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,35,28,8,22,19,31,37,15,44)(2,36,27,7,21,20,32,38,16,43)(3,34,26,5,24,18,29,40,13,41)(4,33,25,6,23,17,30,39,14,42)(11,12), (1,14,3,16)(2,15,4,13)(5,11,7,9)(6,10,8,12)(17,43,19,41)(18,42,20,44)(21,37,23,39)(22,40,24,38)(25,33,27,35)(26,36,28,34)(29,32,31,30)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
5:  $C_5$
6:  $S_3$
10:  $C_{10}$
24:  $S_4$
30:  $S_3 \times C_5$
110:  $F_{11}$
120:  20T34

Resolvents shown for degrees $\leq 29$

Subfields

Degree 2: None

Degree 4: $S_4$

Degree 11: $F_{11}$

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 $ $10$ $11$ $( 1,37,31,22,15, 8,44,35,28,19, 9)( 2,38,32,21,16, 7,43,36,27,20,10) ( 3,39,29,24,13, 6,42,33,26,17,11)( 4,40,30,23,14, 5,41,34,25,18,12)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ $11$ $5$ $( 5,40,18,14,23)( 6,39,17,13,24)( 7,38,20,16,21)( 8,37,19,15,22) ( 9,31,35,28,44)(10,32,36,27,43)(11,29,33,26,42)(12,30,34,25,41)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ $11$ $5$ $( 5,18,23,40,14)( 6,17,24,39,13)( 7,20,21,38,16)( 8,19,22,37,15) ( 9,35,44,31,28)(10,36,43,32,27)(11,33,42,29,26)(12,34,41,30,25)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ $11$ $5$ $( 5,23,14,18,40)( 6,24,13,17,39)( 7,21,16,20,38)( 8,22,15,19,37) ( 9,44,28,35,31)(10,43,27,36,32)(11,42,26,33,29)(12,41,25,34,30)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ $11$ $5$ $( 5,14,40,23,18)( 6,13,39,24,17)( 7,16,38,21,20)( 8,15,37,22,19) ( 9,28,31,44,35)(10,27,32,43,36)(11,26,29,42,33)(12,25,30,41,34)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $3$ $2$ $( 1, 4)( 2, 3)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24) (22,23)(25,28)(26,27)(29,32)(30,31)(33,36)(34,35)(37,40)(38,39)(41,44)(42,43)$
$ 22, 22 $ $30$ $22$ $( 1,40,31,23,15, 5,44,34,28,18, 9, 4,37,30,22,14, 8,41,35,25,19,12) ( 2,39,32,24,16, 6,43,33,27,17,10, 3,38,29,21,13, 7,42,36,26,20,11)$
$ 10, 10, 10, 10, 2, 2 $ $33$ $10$ $( 1, 4)( 2, 3)( 5,37,18,15,23, 8,40,19,14,22)( 6,38,17,16,24, 7,39,20,13,21) ( 9,30,35,25,44,12,31,34,28,41)(10,29,36,26,43,11,32,33,27,42)$
$ 10, 10, 10, 10, 2, 2 $ $33$ $10$ $( 1, 4)( 2, 3)( 5,19,23,37,14, 8,18,22,40,15)( 6,20,24,38,13, 7,17,21,39,16) ( 9,34,44,30,28,12,35,41,31,25)(10,33,43,29,27,11,36,42,32,26)$
$ 10, 10, 10, 10, 2, 2 $ $33$ $10$ $( 1, 4)( 2, 3)( 5,22,14,19,40, 8,23,15,18,37)( 6,21,13,20,39, 7,24,16,17,38) ( 9,41,28,34,31,12,44,25,35,30)(10,42,27,33,32,11,43,26,36,29)$
$ 10, 10, 10, 10, 2, 2 $ $33$ $10$ $( 1, 4)( 2, 3)( 5,15,40,22,18, 8,14,37,23,19)( 6,16,39,21,17, 7,13,38,24,20) ( 9,25,31,41,35,12,28,30,44,34)(10,26,32,42,36,11,27,29,43,33)$
$ 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ $8$ $3$ $( 2, 3, 4)( 5, 7, 6)(10,11,12)(13,14,16)(17,18,20)(21,24,23)(25,27,26) (29,30,32)(33,34,36)(38,39,40)(41,43,42)$
$ 33, 11 $ $40$ $33$ $( 1,37,31,22,15, 8,44,35,28,19, 9)( 2,39,30,21,13, 5,43,33,25,20,11, 4,38,29, 23,16, 6,41,36,26,18,10, 3,40,32,24,14, 7,42,34,27,17,12)$
$ 33, 11 $ $40$ $33$ $( 1,31,15,44,28, 9,37,22, 8,35,19)( 2,29,14,43,26,12,38,24, 5,36,17, 4,32,13, 41,27,11,40,21, 6,34,20, 3,30,16,42,25,10,39,23, 7,33,18)$
$ 15, 15, 5, 5, 3, 1 $ $88$ $15$ $( 2, 3, 4)( 5,38,17,14,21, 6,40,20,13,23, 7,39,18,16,24)( 8,37,19,15,22) ( 9,31,35,28,44)(10,29,34,27,42,12,32,33,25,43,11,30,36,26,41)$
$ 15, 15, 5, 5, 3, 1 $ $88$ $15$ $( 2, 3, 4)( 5,20,24,40,16, 6,18,21,39,14, 7,17,23,38,13)( 8,19,22,37,15) ( 9,35,44,31,28)(10,33,41,32,26,12,36,42,30,27,11,34,43,29,25)$
$ 15, 15, 5, 5, 3, 1 $ $88$ $15$ $( 2, 3, 4)( 5,21,13,18,38, 6,23,16,17,40, 7,24,14,20,39)( 8,22,15,19,37) ( 9,44,28,35,31)(10,42,25,36,29,12,43,26,34,32,11,41,27,33,30)$
$ 15, 15, 5, 5, 3, 1 $ $88$ $15$ $( 2, 3, 4)( 5,16,39,23,20, 6,14,38,24,18, 7,13,40,21,17)( 8,15,37,22,19) ( 9,28,31,44,35)(10,26,30,43,33,12,27,29,41,36,11,25,32,42,34)$
$ 10, 10, 10, 10, 2, 1, 1 $ $66$ $10$ $( 3, 4)( 5,33,40,26,18,42,14,11,23,29)( 6,34,39,25,17,41,13,12,24,30) ( 7,36,38,27,20,43,16,10,21,32)( 8,35,37,28,19,44,15, 9,22,31)$
$ 10, 10, 10, 10, 2, 1, 1 $ $66$ $10$ $( 3, 4)( 5,26,14,29,40,42,23,33,18,11)( 6,25,13,30,39,41,24,34,17,12) ( 7,27,16,32,38,43,21,36,20,10)( 8,28,15,31,37,44,22,35,19, 9)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1 $ $66$ $2$ $( 3, 4)( 5,42)( 6,41)( 7,43)( 8,44)( 9,37)(10,38)(11,40)(12,39)(13,34)(14,33) (15,35)(16,36)(17,30)(18,29)(19,31)(20,32)(21,27)(22,28)(23,26)(24,25)$
$ 10, 10, 10, 10, 2, 1, 1 $ $66$ $10$ $( 3, 4)( 5,29,23,11,14,42,18,26,40,33)( 6,30,24,12,13,41,17,25,39,34) ( 7,32,21,10,16,43,20,27,38,36)( 8,31,22, 9,15,44,19,28,37,35)$
$ 10, 10, 10, 10, 2, 1, 1 $ $66$ $10$ $( 3, 4)( 5,11,18,33,23,42,40,29,14,26)( 6,12,17,34,24,41,39,30,13,25) ( 7,10,20,36,21,43,38,32,16,27)( 8, 9,19,35,22,44,37,31,15,28)$
$ 20, 20, 4 $ $66$ $20$ $( 1, 4, 2, 3)( 5,36,39,28,18,43,13, 9,23,32, 6,35,40,27,17,44,14,10,24,31) ( 7,33,37,25,20,42,15,12,21,29, 8,34,38,26,19,41,16,11,22,30)$
$ 20, 20, 4 $ $66$ $20$ $( 1, 4, 2, 3)( 5,27,13,31,40,43,24,35,18,10, 6,28,14,32,39,44,23,36,17, 9) ( 7,26,15,30,38,42,22,34,20,11, 8,25,16,29,37,41,21,33,19,12)$
$ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ $66$ $4$ $( 1, 4, 2, 3)( 5,43, 6,44)( 7,42, 8,41)( 9,40,10,39)(11,37,12,38)(13,35,14,36) (15,34,16,33)(17,31,18,32)(19,30,20,29)(21,26,22,25)(23,27,24,28)$
$ 20, 20, 4 $ $66$ $20$ $( 1, 4, 2, 3)( 5,32,24, 9,14,43,17,28,40,36, 6,31,23,10,13,44,18,27,39,35) ( 7,29,22,12,16,42,19,25,38,33, 8,30,21,11,15,41,20,26,37,34)$
$ 20, 20, 4 $ $66$ $20$ $( 1, 4, 2, 3)( 5,10,17,35,23,43,39,31,14,27, 6, 9,18,36,24,44,40,32,13,28) ( 7,11,19,34,21,42,37,30,16,26, 8,12,20,33,22,41,38,29,15,25)$

Group invariants

Order:  $1320=2^{3} \cdot 3 \cdot 5 \cdot 11$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [1320, 140]
Character table: Data not available.