# Properties

 Label 44T4 Order $$44$$ n $$44$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $D_{22}$

## Group action invariants

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

## Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$ x 3
4:  $C_2^2$
22:  $D_{11}$

Resolvents shown for degrees $\leq 29$

## Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$

Degree 11: $D_{11}$

Degree 22: 22T2, $D_{22}$ x 2

## Low degree siblings

22T3 x 2

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

## Group invariants

 Order: $44=2^{2} \cdot 11$ Cyclic: No Abelian: No Solvable: Yes GAP id: [44, 3]
 Character table:  2 2 2 2 2 1 1 1 1 1 1 1 1 1 1 11 1 1 . . 1 1 1 1 1 1 1 1 1 1 1a 2a 2b 2c 22a 11a 22b 11b 11c 22c 22d 11d 22e 11e 2P 1a 1a 1a 1a 11b 11b 11d 11d 11e 11e 11c 11c 11a 11a 3P 1a 2a 2b 2c 22c 11c 22e 11e 11b 22b 22a 11a 22d 11d 5P 1a 2a 2b 2c 22e 11e 22a 11a 11d 22d 22b 11b 22c 11c 7P 1a 2a 2b 2c 22d 11d 22c 11c 11a 22a 22e 11e 22b 11b 11P 1a 2a 2b 2c 2a 1a 2a 1a 1a 2a 2a 1a 2a 1a 13P 1a 2a 2b 2c 22b 11b 22d 11d 11e 22e 22c 11c 22a 11a 17P 1a 2a 2b 2c 22e 11e 22a 11a 11d 22d 22b 11b 22c 11c 19P 1a 2a 2b 2c 22c 11c 22e 11e 11b 22b 22a 11a 22d 11d X.1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 X.2 1 -1 -1 1 -1 1 -1 1 1 -1 -1 1 -1 1 X.3 1 -1 1 -1 -1 1 -1 1 1 -1 -1 1 -1 1 X.4 1 1 -1 -1 1 1 1 1 1 1 1 1 1 1 X.5 2 2 . . A A D D E E B B C C X.6 2 2 . . B B E E A A C C D D X.7 2 2 . . C C A A B B D D E E X.8 2 2 . . D D B B C C E E A A X.9 2 2 . . E E C C D D A A B B X.10 2 -2 . . -A A -D D E -E -B B -C C X.11 2 -2 . . -B B -E E A -A -C C -D D X.12 2 -2 . . -C C -A A B -B -D D -E E X.13 2 -2 . . -D D -B B C -C -E E -A A X.14 2 -2 . . -E E -C C D -D -A A -B B A = E(11)^4+E(11)^7 B = E(11)^5+E(11)^6 C = E(11)^2+E(11)^9 D = E(11)^3+E(11)^8 E = E(11)+E(11)^10