Group action invariants
| Degree $n$: | $44$ | |
| Transitive number $t$: | $13$ | |
| Group: | $C_{11}:C_{20}$ | |
| Parity: | $-1$ | |
| Primitive: | no | |
| Nilpotency class: | $-1$ (not nilpotent) | |
| $|\Aut(F/K)|$: | $4$ | |
| Generators: | (1,3,2,4)(5,33,38,26,17,42,14,10,21,29,6,34,37,25,18,41,13,9,22,30)(7,36,39,27,19,43,15,12,23,32,8,35,40,28,20,44,16,11,24,31), (1,39,7,9,26,2,40,8,10,25)(3,38,5,12,28,4,37,6,11,27)(13,44,21,31,17,14,43,22,32,18)(15,41,24,30,20,16,42,23,29,19)(33,34)(35,36) |
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ $4$: $C_4$ $5$: $C_5$ $10$: $C_{10}$ $20$: 20T1 $110$: $F_{11}$ Resolvents shown for degrees $\leq 29$
Subfields
Degree 2: $C_2$
Degree 4: $C_4$
Degree 11: $F_{11}$
Degree 22: 22T4
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 $ | $1$ | $1$ | $()$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ | $11$ | $5$ | $( 5,13,37,21,17)( 6,14,38,22,18)( 7,16,40,23,19)( 8,15,39,24,20) ( 9,25,29,42,33)(10,26,30,41,34)(11,28,32,43,36)(12,27,31,44,35)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ | $11$ | $5$ | $( 5,17,21,37,13)( 6,18,22,38,14)( 7,19,23,40,16)( 8,20,24,39,15) ( 9,33,42,29,25)(10,34,41,30,26)(11,36,43,32,28)(12,35,44,31,27)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ | $11$ | $5$ | $( 5,21,13,17,37)( 6,22,14,18,38)( 7,23,16,19,40)( 8,24,15,20,39) ( 9,42,25,33,29)(10,41,26,34,30)(11,43,28,36,32)(12,44,27,35,31)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ | $11$ | $5$ | $( 5,37,17,13,21)( 6,38,18,14,22)( 7,40,19,16,23)( 8,39,20,15,24) ( 9,29,33,25,42)(10,30,34,26,41)(11,32,36,28,43)(12,31,35,27,44)$ |
| $ 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 $ | $11$ | $10$ | $( 1, 2)( 3, 4)( 5,14,37,22,17, 6,13,38,21,18)( 7,15,40,24,19, 8,16,39,23,20) ( 9,26,29,41,33,10,25,30,42,34)(11,27,32,44,36,12,28,31,43,35)$ |
| $ 10, 10, 10, 10, 2, 2 $ | $11$ | $10$ | $( 1, 2)( 3, 4)( 5,18,21,38,13, 6,17,22,37,14)( 7,20,23,39,16, 8,19,24,40,15) ( 9,34,42,30,25,10,33,41,29,26)(11,35,43,31,28,12,36,44,32,27)$ |
| $ 10, 10, 10, 10, 2, 2 $ | $11$ | $10$ | $( 1, 2)( 3, 4)( 5,22,13,18,37, 6,21,14,17,38)( 7,24,16,20,40, 8,23,15,19,39) ( 9,41,25,34,29,10,42,26,33,30)(11,44,28,35,32,12,43,27,36,31)$ |
| $ 10, 10, 10, 10, 2, 2 $ | $11$ | $10$ | $( 1, 2)( 3, 4)( 5,38,17,14,21, 6,37,18,13,22)( 7,39,19,15,23, 8,40,20,16,24) ( 9,30,33,26,42,10,29,34,25,41)(11,31,36,27,43,12,32,35,28,44)$ |
| $ 20, 20, 4 $ | $11$ | $20$ | $( 1, 3, 2, 4)( 5, 9,18,34,21,42,38,30,13,25, 6,10,17,33,22,41,37,29,14,26) ( 7,11,20,35,23,43,39,31,16,28, 8,12,19,36,24,44,40,32,15,27)$ |
| $ 20, 20, 4 $ | $11$ | $20$ | $( 1, 3, 2, 4)( 5,25,14,30,37,42,22,34,17, 9, 6,26,13,29,38,41,21,33,18,10) ( 7,28,15,31,40,43,24,35,19,11, 8,27,16,32,39,44,23,36,20,12)$ |
| $ 20, 20, 4 $ | $11$ | $20$ | $( 1, 3, 2, 4)( 5,29,22,10,13,42,18,26,37,33, 6,30,21, 9,14,41,17,25,38,34) ( 7,32,24,12,16,43,20,27,40,36, 8,31,23,11,15,44,19,28,39,35)$ |
| $ 20, 20, 4 $ | $11$ | $20$ | $( 1, 3, 2, 4)( 5,33,38,26,17,42,14,10,21,29, 6,34,37,25,18,41,13, 9,22,30) ( 7,36,39,27,19,43,15,12,23,32, 8,35,40,28,20,44,16,11,24,31)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $11$ | $4$ | $( 1, 3, 2, 4)( 5,42, 6,41)( 7,43, 8,44)( 9,38,10,37)(11,39,12,40)(13,33,14,34) (15,35,16,36)(17,29,18,30)(19,32,20,31)(21,25,22,26)(23,28,24,27)$ |
| $ 20, 20, 4 $ | $11$ | $20$ | $( 1, 4, 2, 3)( 5,10,18,33,21,41,38,29,13,26, 6, 9,17,34,22,42,37,30,14,25) ( 7,12,20,36,23,44,39,32,16,27, 8,11,19,35,24,43,40,31,15,28)$ |
| $ 20, 20, 4 $ | $11$ | $20$ | $( 1, 4, 2, 3)( 5,26,14,29,37,41,22,33,17,10, 6,25,13,30,38,42,21,34,18, 9) ( 7,27,15,32,40,44,24,36,19,12, 8,28,16,31,39,43,23,35,20,11)$ |
| $ 20, 20, 4 $ | $11$ | $20$ | $( 1, 4, 2, 3)( 5,30,22, 9,13,41,18,25,37,34, 6,29,21,10,14,42,17,26,38,33) ( 7,31,24,11,16,44,20,28,40,35, 8,32,23,12,15,43,19,27,39,36)$ |
| $ 20, 20, 4 $ | $11$ | $20$ | $( 1, 4, 2, 3)( 5,34,38,25,17,41,14, 9,21,30, 6,33,37,26,18,42,13,10,22,29) ( 7,35,39,28,19,44,15,11,23,31, 8,36,40,27,20,43,16,12,24,32)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $11$ | $4$ | $( 1, 4, 2, 3)( 5,41, 6,42)( 7,44, 8,43)( 9,37,10,38)(11,40,12,39)(13,34,14,33) (15,36,16,35)(17,30,18,29)(19,31,20,32)(21,26,22,25)(23,27,24,28)$ |
| $ 11, 11, 11, 11 $ | $10$ | $11$ | $( 1, 7,10,16,19,23,26,30,34,40,41)( 2, 8, 9,15,20,24,25,29,33,39,42) ( 3, 5,11,13,17,21,28,32,36,37,43)( 4, 6,12,14,18,22,27,31,35,38,44)$ |
| $ 22, 22 $ | $10$ | $22$ | $( 1, 8,10,15,19,24,26,29,34,39,41, 2, 7, 9,16,20,23,25,30,33,40,42) ( 3, 6,11,14,17,22,28,31,36,38,43, 4, 5,12,13,18,21,27,32,35,37,44)$ |
Group invariants
| Order: | $220=2^{2} \cdot 5 \cdot 11$ | |
| Cyclic: | no | |
| Abelian: | no | |
| Solvable: | yes | |
| GAP id: | [220, 1] |
| Character table: not available. |