Properties

 Label 44T3 Order $$44$$ n $$44$$ Cyclic No Abelian No Solvable Yes Primitive No $p$-group No Group: $C_{11}:C_4$

Group action invariants

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

Low degree resolvents

|G/N|Galois groups for stem field(s)
2:  $C_2$
4:  $C_4$
22:  $D_{11}$

Resolvents shown for degrees $\leq 29$

Subfields

Degree 2: $C_2$

Degree 4: $C_4$

Degree 11: $D_{11}$

Degree 22: 22T2

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

Group invariants

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