Group action invariants
| Degree $n$: | $44$ | |
| Transitive number $t$: | $3$ | |
| Group: | $C_{11}:C_4$ | |
| Parity: | $-1$ | |
| Primitive: | no | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $44$ | |
| 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) |
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
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