Properties

Label 44T16
Order \(220\)
n \(44\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^2\times C_{11}:C_5$

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Group action invariants

Degree $n$ :  $44$
Transitive number $t$ :  $16$
Group :  $C_2^2\times C_{11}:C_5$
Parity:  $1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,16,20,36,11,3,13,18,34,10)(2,15,19,35,12,4,14,17,33,9)(5,32,37,28,21,8,30,40,26,23)(6,31,38,27,22,7,29,39,25,24)(41,43)(42,44), (1,9,42,40,22,4,11,43,38,23)(2,10,41,39,21,3,12,44,37,24)(5,27,19,31,33,7,26,18,30,36)(6,28,20,32,34,8,25,17,29,35)(13,15)(14,16)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
5:  $C_5$
10:  $C_{10}$ x 3
20:  20T3
55:  $C_{11}:C_5$
110:  22T5 x 3

Resolvents shown for degrees $\leq 29$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 11: $C_{11}:C_5$

Degree 22: 22T5 x 3

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

Group invariants

Order:  $220=2^{2} \cdot 5 \cdot 11$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [220, 8]
Character table: Data not available.