Properties

Label 44T30
Order \(528\)
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 $t$ :  $30$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,28,2,27)(3,25,4,26)(5,22,6,21)(7,23,8,24)(9,19,10,20)(11,18,12,17)(13,15,14,16)(29,44,30,43)(31,41,32,42)(33,39,34,40)(35,38,36,37), (1,9,4,10,2,11)(3,12)(5,8,7)(13,42,16,41,14,44)(15,43)(17,40,18,38,19,37)(20,39)(21,36,22,34,23,33)(24,35)(25,31,26,29,27,30)(28,32)
$|\Aut(F/K)|$:  $1$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
6:  $S_3$
12:  $D_{6}$
22:  $D_{11}$
24:  $S_4$
44:  $D_{22}$
48:  $S_4\times C_2$

Resolvents shown for degrees $\leq 29$

Subfields

Degree 2: None

Degree 4: $S_4$

Degree 11: $D_{11}$

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

Group invariants

Order:  $528=2^{4} \cdot 3 \cdot 11$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [528, 160]
Character table: Data not available.