Properties

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

Learn more about

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 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 $ $1$ $1$ $()$
$ 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