Properties

Label 44T25
Order \(440\)
n \(44\)
Cyclic No
Abelian No
Solvable Yes
Primitive No
$p$-group No
Group: $C_2^2\times F_{11}$

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Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 7
4:  $C_2^2$ x 7
5:  $C_5$
8:  $C_2^3$
10:  $C_{10}$ x 7
20:  20T3 x 7
110:  $F_{11}$
220:  22T6 x 3

Resolvents shown for degrees $\leq 29$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 11: $F_{11}$

Degree 22: 22T6 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 $ $1$ $1$ $()$
$ 10, 10, 10, 10, 1, 1, 1, 1 $ $11$ $10$ $( 5,11,19,34,22,42,37,29,13,25)( 6,12,20,33,21,41,38,30,14,26)( 7,10,18,35,23, 43,40,31,16,27)( 8, 9,17,36,24,44,39,32,15,28)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ $11$ $5$ $( 5,13,37,22,19)( 6,14,38,21,20)( 7,16,40,23,18)( 8,15,39,24,17) ( 9,28,32,44,36)(10,27,31,43,35)(11,25,29,42,34)(12,26,30,41,33)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ $11$ $5$ $( 5,19,22,37,13)( 6,20,21,38,14)( 7,18,23,40,16)( 8,17,24,39,15) ( 9,36,44,32,28)(10,35,43,31,27)(11,34,42,29,25)(12,33,41,30,26)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ $11$ $5$ $( 5,22,13,19,37)( 6,21,14,20,38)( 7,23,16,18,40)( 8,24,15,17,39) ( 9,44,28,36,32)(10,43,27,35,31)(11,42,25,34,29)(12,41,26,33,30)$
$ 10, 10, 10, 10, 1, 1, 1, 1 $ $11$ $10$ $( 5,25,13,29,37,42,22,34,19,11)( 6,26,14,30,38,41,21,33,20,12)( 7,27,16,31,40, 43,23,35,18,10)( 8,28,15,32,39,44,24,36,17, 9)$
$ 10, 10, 10, 10, 1, 1, 1, 1 $ $11$ $10$ $( 5,29,22,11,13,42,19,25,37,34)( 6,30,21,12,14,41,20,26,38,33)( 7,31,23,10,16, 43,18,27,40,35)( 8,32,24, 9,15,44,17,28,39,36)$
$ 10, 10, 10, 10, 1, 1, 1, 1 $ $11$ $10$ $( 5,34,37,25,19,42,13,11,22,29)( 6,33,38,26,20,41,14,12,21,30)( 7,35,40,27,18, 43,16,10,23,31)( 8,36,39,28,17,44,15, 9,24,32)$
$ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ $11$ $5$ $( 5,37,19,13,22)( 6,38,20,14,21)( 7,40,18,16,23)( 8,39,17,15,24) ( 9,32,36,28,44)(10,31,35,27,43)(11,29,34,25,42)(12,30,33,26,41)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ $11$ $2$ $( 5,42)( 6,41)( 7,43)( 8,44)( 9,39)(10,40)(11,37)(12,38)(13,34)(14,33)(15,36) (16,35)(17,32)(18,31)(19,29)(20,30)(21,26)(22,25)(23,27)(24,28)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,24)(25,26)(27,28)(29,30)(31,32)(33,34)(35,36)(37,38)(39,40)(41,42)(43,44)$
$ 10, 10, 10, 10, 2, 2 $ $11$ $10$ $( 1, 2)( 3, 4)( 5,12,19,33,22,41,37,30,13,26)( 6,11,20,34,21,42,38,29,14,25) ( 7, 9,18,36,23,44,40,32,16,28)( 8,10,17,35,24,43,39,31,15,27)$
$ 10, 10, 10, 10, 2, 2 $ $11$ $10$ $( 1, 2)( 3, 4)( 5,14,37,21,19, 6,13,38,22,20)( 7,15,40,24,18, 8,16,39,23,17) ( 9,27,32,43,36,10,28,31,44,35)(11,26,29,41,34,12,25,30,42,33)$
$ 10, 10, 10, 10, 2, 2 $ $11$ $10$ $( 1, 2)( 3, 4)( 5,20,22,38,13, 6,19,21,37,14)( 7,17,23,39,16, 8,18,24,40,15) ( 9,35,44,31,28,10,36,43,32,27)(11,33,42,30,25,12,34,41,29,26)$
$ 10, 10, 10, 10, 2, 2 $ $11$ $10$ $( 1, 2)( 3, 4)( 5,21,13,20,37, 6,22,14,19,38)( 7,24,16,17,40, 8,23,15,18,39) ( 9,43,28,35,32,10,44,27,36,31)(11,41,25,33,29,12,42,26,34,30)$
$ 10, 10, 10, 10, 2, 2 $ $11$ $10$ $( 1, 2)( 3, 4)( 5,26,13,30,37,41,22,33,19,12)( 6,25,14,29,38,42,21,34,20,11) ( 7,28,16,32,40,44,23,36,18, 9)( 8,27,15,31,39,43,24,35,17,10)$
$ 10, 10, 10, 10, 2, 2 $ $11$ $10$ $( 1, 2)( 3, 4)( 5,30,22,12,13,41,19,26,37,33)( 6,29,21,11,14,42,20,25,38,34) ( 7,32,23, 9,16,44,18,28,40,36)( 8,31,24,10,15,43,17,27,39,35)$
$ 10, 10, 10, 10, 2, 2 $ $11$ $10$ $( 1, 2)( 3, 4)( 5,33,37,26,19,41,13,12,22,30)( 6,34,38,25,20,42,14,11,21,29) ( 7,36,40,28,18,44,16, 9,23,32)( 8,35,39,27,17,43,15,10,24,31)$
$ 10, 10, 10, 10, 2, 2 $ $11$ $10$ $( 1, 2)( 3, 4)( 5,38,19,14,22, 6,37,20,13,21)( 7,39,18,15,23, 8,40,17,16,24) ( 9,31,36,27,44,10,32,35,28,43)(11,30,34,26,42,12,29,33,25,41)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $11$ $2$ $( 1, 2)( 3, 4)( 5,41)( 6,42)( 7,44)( 8,43)( 9,40)(10,39)(11,38)(12,37)(13,33) (14,34)(15,35)(16,36)(17,31)(18,32)(19,30)(20,29)(21,25)(22,26)(23,28)(24,27)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23) (22,24)(25,28)(26,27)(29,32)(30,31)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)$
$ 10, 10, 10, 10, 2, 2 $ $11$ $10$ $( 1, 3)( 2, 4)( 5, 9,19,36,22,44,37,32,13,28)( 6,10,20,35,21,43,38,31,14,27) ( 7,12,18,33,23,41,40,30,16,26)( 8,11,17,34,24,42,39,29,15,25)$
$ 10, 10, 10, 10, 2, 2 $ $11$ $10$ $( 1, 3)( 2, 4)( 5,15,37,24,19, 8,13,39,22,17)( 6,16,38,23,20, 7,14,40,21,18) ( 9,25,32,42,36,11,28,29,44,34)(10,26,31,41,35,12,27,30,43,33)$
$ 10, 10, 10, 10, 2, 2 $ $11$ $10$ $( 1, 3)( 2, 4)( 5,17,22,39,13, 8,19,24,37,15)( 6,18,21,40,14, 7,20,23,38,16) ( 9,34,44,29,28,11,36,42,32,25)(10,33,43,30,27,12,35,41,31,26)$
$ 10, 10, 10, 10, 2, 2 $ $11$ $10$ $( 1, 3)( 2, 4)( 5,24,13,17,37, 8,22,15,19,39)( 6,23,14,18,38, 7,21,16,20,40) ( 9,42,28,34,32,11,44,25,36,29)(10,41,27,33,31,12,43,26,35,30)$
$ 10, 10, 10, 10, 2, 2 $ $11$ $10$ $( 1, 3)( 2, 4)( 5,28,13,32,37,44,22,36,19, 9)( 6,27,14,31,38,43,21,35,20,10) ( 7,26,16,30,40,41,23,33,18,12)( 8,25,15,29,39,42,24,34,17,11)$
$ 10, 10, 10, 10, 2, 2 $ $11$ $10$ $( 1, 3)( 2, 4)( 5,32,22, 9,13,44,19,28,37,36)( 6,31,21,10,14,43,20,27,38,35) ( 7,30,23,12,16,41,18,26,40,33)( 8,29,24,11,15,42,17,25,39,34)$
$ 10, 10, 10, 10, 2, 2 $ $11$ $10$ $( 1, 3)( 2, 4)( 5,36,37,28,19,44,13, 9,22,32)( 6,35,38,27,20,43,14,10,21,31) ( 7,33,40,26,18,41,16,12,23,30)( 8,34,39,25,17,42,15,11,24,29)$
$ 10, 10, 10, 10, 2, 2 $ $11$ $10$ $( 1, 3)( 2, 4)( 5,39,19,15,22, 8,37,17,13,24)( 6,40,20,16,21, 7,38,18,14,23) ( 9,29,36,25,44,11,32,34,28,42)(10,30,35,26,43,12,31,33,27,41)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $11$ $2$ $( 1, 3)( 2, 4)( 5,44)( 6,43)( 7,41)( 8,42)( 9,37)(10,38)(11,39)(12,40)(13,36) (14,35)(15,34)(16,33)(17,29)(18,30)(19,32)(20,31)(21,27)(22,28)(23,26)(24,25)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $1$ $2$ $( 1, 4)( 2, 3)( 5, 7)( 6, 8)( 9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,24) (22,23)(25,27)(26,28)(29,31)(30,32)(33,36)(34,35)(37,40)(38,39)(41,44)(42,43)$
$ 10, 10, 10, 10, 2, 2 $ $11$ $10$ $( 1, 4)( 2, 3)( 5,10,19,35,22,43,37,31,13,27)( 6, 9,20,36,21,44,38,32,14,28) ( 7,11,18,34,23,42,40,29,16,25)( 8,12,17,33,24,41,39,30,15,26)$
$ 10, 10, 10, 10, 2, 2 $ $11$ $10$ $( 1, 4)( 2, 3)( 5,16,37,23,19, 7,13,40,22,18)( 6,15,38,24,20, 8,14,39,21,17) ( 9,26,32,41,36,12,28,30,44,33)(10,25,31,42,35,11,27,29,43,34)$
$ 10, 10, 10, 10, 2, 2 $ $11$ $10$ $( 1, 4)( 2, 3)( 5,18,22,40,13, 7,19,23,37,16)( 6,17,21,39,14, 8,20,24,38,15) ( 9,33,44,30,28,12,36,41,32,26)(10,34,43,29,27,11,35,42,31,25)$
$ 10, 10, 10, 10, 2, 2 $ $11$ $10$ $( 1, 4)( 2, 3)( 5,23,13,18,37, 7,22,16,19,40)( 6,24,14,17,38, 8,21,15,20,39) ( 9,41,28,33,32,12,44,26,36,30)(10,42,27,34,31,11,43,25,35,29)$
$ 10, 10, 10, 10, 2, 2 $ $11$ $10$ $( 1, 4)( 2, 3)( 5,27,13,31,37,43,22,35,19,10)( 6,28,14,32,38,44,21,36,20, 9) ( 7,25,16,29,40,42,23,34,18,11)( 8,26,15,30,39,41,24,33,17,12)$
$ 10, 10, 10, 10, 2, 2 $ $11$ $10$ $( 1, 4)( 2, 3)( 5,31,22,10,13,43,19,27,37,35)( 6,32,21, 9,14,44,20,28,38,36) ( 7,29,23,11,16,42,18,25,40,34)( 8,30,24,12,15,41,17,26,39,33)$
$ 10, 10, 10, 10, 2, 2 $ $11$ $10$ $( 1, 4)( 2, 3)( 5,35,37,27,19,43,13,10,22,31)( 6,36,38,28,20,44,14, 9,21,32) ( 7,34,40,25,18,42,16,11,23,29)( 8,33,39,26,17,41,15,12,24,30)$
$ 10, 10, 10, 10, 2, 2 $ $11$ $10$ $( 1, 4)( 2, 3)( 5,40,19,16,22, 7,37,18,13,23)( 6,39,20,15,21, 8,38,17,14,24) ( 9,30,36,26,44,12,32,33,28,41)(10,29,35,25,43,11,31,34,27,42)$
$ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ $11$ $2$ $( 1, 4)( 2, 3)( 5,43)( 6,44)( 7,42)( 8,41)( 9,38)(10,37)(11,40)(12,39)(13,35) (14,36)(15,33)(16,34)(17,30)(18,29)(19,31)(20,32)(21,28)(22,27)(23,25)(24,26)$
$ 22, 22 $ $10$ $22$ $( 1, 5,12,13,20,22,26,29,33,37,41, 2, 6,11,14,19,21,25,30,34,38,42) ( 3, 8,10,15,18,24,27,32,35,39,43, 4, 7, 9,16,17,23,28,31,36,40,44)$
$ 11, 11, 11, 11 $ $10$ $11$ $( 1, 6,12,14,20,21,26,30,33,38,41)( 2, 5,11,13,19,22,25,29,34,37,42) ( 3, 7,10,16,18,23,27,31,35,40,43)( 4, 8, 9,15,17,24,28,32,36,39,44)$
$ 22, 22 $ $10$ $22$ $( 1, 7,12,16,20,23,26,31,33,40,41, 3, 6,10,14,18,21,27,30,35,38,43) ( 2, 8,11,15,19,24,25,32,34,39,42, 4, 5, 9,13,17,22,28,29,36,37,44)$
$ 22, 22 $ $10$ $22$ $( 1, 8,12,15,20,24,26,32,33,39,41, 4, 6, 9,14,17,21,28,30,36,38,44) ( 2, 7,11,16,19,23,25,31,34,40,42, 3, 5,10,13,18,22,27,29,35,37,43)$

Group invariants

Order:  $440=2^{3} \cdot 5 \cdot 11$
Cyclic:  No
Abelian:  No
Solvable:  Yes
GAP id:  [440, 42]
Character table: Data not available.