Properties

Label 44T15
Order \(220\)
n \(44\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_4\times C_{11}:C_5$

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Group action invariants

Degree $n$ :  $44$
Transitive number $t$ :  $15$
Group :  $C_4\times C_{11}:C_5$
Parity:  $-1$
Primitive:  No
Nilpotency class:  $-1$ (not nilpotent)
Generators:  (1,31,12,18,6,3,29,9,19,8,2,32,11,17,5,4,30,10,20,7)(13,35,26,40,42,16,33,27,37,44,14,36,25,39,41,15,34,28,38,43)(21,23,22,24), (1,23,26,18,34,3,22,27,19,35,2,24,25,17,33,4,21,28,20,36)(5,15,42,31,12,7,13,44,29,9,6,16,41,32,11,8,14,43,30,10)(37,39,38,40)
$|\Aut(F/K)|$:  $4$

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
5:  $C_5$
10:  $C_{10}$
20:  20T1
55:  $C_{11}:C_5$
110:  22T5

Resolvents shown for degrees $\leq 29$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 11: $C_{11}:C_5$

Degree 22: 22T5

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

Group invariants

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