Show commands:
Magma
magma: G := TransitiveGroup(44, 40);
Group action invariants
| Degree $n$: | $44$ | magma: t, n := TransitiveGroupIdentification(G); n;
| |
| Transitive number $t$: | $40$ | magma: t, n := TransitiveGroupIdentification(G); t;
| |
| Group: | $D_{11}\wr C_2$ | ||
| Parity: | $1$ | magma: IsEven(G);
| |
| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
|
| $\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
| |
| Generators: | (1,40,17,31,11,23,6,37,22,29,15,44,9,35,4,27,20,42,13,34,7,26)(2,39,18,32,12,24,5,38,21,30,16,43,10,36,3,28,19,41,14,33,8,25), (1,24,9,41)(2,23,10,42)(3,34,8,31)(4,33,7,32)(5,44)(6,43)(11,30,22,36)(12,29,21,35)(13,39,20,25)(14,40,19,26)(15,28,17,38)(16,27,18,37) | magma: Generators(G);
|
Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $8$: $D_{4}$ Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$
Degree 11: None
Degree 22: 22T10
Low degree siblings
22T10 x 2, 44T39 x 2, 44T40, 44T41 x 2, 44T44 x 2Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
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 $ | $1$ | $1$ | $()$ |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1,17,11, 6,22,15, 9, 4,20,13, 7)( 2,18,12, 5,21,16,10, 3,19,14, 8) (23,37,29,44,35,27,42,34,26,40,31)(24,38,30,43,36,28,41,33,25,39,32)$ |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1,11,22, 9,20, 7,17, 6,15, 4,13)( 2,12,21,10,19, 8,18, 5,16, 3,14) (23,29,35,42,26,31,37,44,27,34,40)(24,30,36,41,25,32,38,43,28,33,39)$ |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1,22,20,17,15,13,11, 9, 7, 6, 4)( 2,21,19,18,16,14,12,10, 8, 5, 3) (23,35,26,37,27,40,29,42,31,44,34)(24,36,25,38,28,39,30,41,32,43,33)$ |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1,20,15,11, 7, 4,22,17,13, 9, 6)( 2,19,16,12, 8, 3,21,18,14,10, 5) (23,26,27,29,31,34,35,37,40,42,44)(24,25,28,30,32,33,36,38,39,41,43)$ |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1,15, 7,22,13, 6,20,11, 4,17, 9)( 2,16, 8,21,14, 5,19,12, 3,18,10) (23,27,31,35,40,44,26,29,34,37,42)(24,28,32,36,39,43,25,30,33,38,41)$ |
| $ 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $11$ | $(23,29,35,42,26,31,37,44,27,34,40)(24,30,36,41,25,32,38,43,28,33,39)$ |
| $ 11, 11, 11, 11 $ | $8$ | $11$ | $( 1,17,11, 6,22,15, 9, 4,20,13, 7)( 2,18,12, 5,21,16,10, 3,19,14, 8) (23,44,42,40,37,35,34,31,29,27,26)(24,43,41,39,38,36,33,32,30,28,25)$ |
| $ 11, 11, 11, 11 $ | $8$ | $11$ | $( 1,11,22, 9,20, 7,17, 6,15, 4,13)( 2,12,21,10,19, 8,18, 5,16, 3,14) (23,35,26,37,27,40,29,42,31,44,34)(24,36,25,38,28,39,30,41,32,43,33)$ |
| $ 11, 11, 11, 11 $ | $8$ | $11$ | $( 1,22,20,17,15,13,11, 9, 7, 6, 4)( 2,21,19,18,16,14,12,10, 8, 5, 3) (23,42,37,34,29,26,44,40,35,31,27)(24,41,38,33,30,25,43,39,36,32,28)$ |
| $ 11, 11, 11, 11 $ | $8$ | $11$ | $( 1, 6, 9,13,17,22, 4, 7,11,15,20)( 2, 5,10,14,18,21, 3, 8,12,16,19) (23,27,31,35,40,44,26,29,34,37,42)(24,28,32,36,39,43,25,30,33,38,41)$ |
| $ 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $11$ | $(23,35,26,37,27,40,29,42,31,44,34)(24,36,25,38,28,39,30,41,32,43,33)$ |
| $ 11, 11, 11, 11 $ | $8$ | $11$ | $( 1,17,11, 6,22,15, 9, 4,20,13, 7)( 2,18,12, 5,21,16,10, 3,19,14, 8) (23,27,31,35,40,44,26,29,34,37,42)(24,28,32,36,39,43,25,30,33,38,41)$ |
| $ 11, 11, 11, 11 $ | $8$ | $11$ | $( 1,11,22, 9,20, 7,17, 6,15, 4,13)( 2,12,21,10,19, 8,18, 5,16, 3,14) (23,42,37,34,29,26,44,40,35,31,27)(24,41,38,33,30,25,43,39,36,32,28)$ |
| $ 11, 11, 11, 11 $ | $8$ | $11$ | $( 1,22,20,17,15,13,11, 9, 7, 6, 4)( 2,21,19,18,16,14,12,10, 8, 5, 3) (23,26,27,29,31,34,35,37,40,42,44)(24,25,28,30,32,33,36,38,39,41,43)$ |
| $ 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $11$ | $(23,26,27,29,31,34,35,37,40,42,44)(24,25,28,30,32,33,36,38,39,41,43)$ |
| $ 11, 11, 11, 11 $ | $8$ | $11$ | $( 1,17,11, 6,22,15, 9, 4,20,13, 7)( 2,18,12, 5,21,16,10, 3,19,14, 8) (23,40,34,27,44,37,31,26,42,35,29)(24,39,33,28,43,38,32,25,41,36,30)$ |
| $ 11, 11, 11, 11 $ | $8$ | $11$ | $( 1,22,20,17,15,13,11, 9, 7, 6, 4)( 2,21,19,18,16,14,12,10, 8, 5, 3) (23,37,29,44,35,27,42,34,26,40,31)(24,38,30,43,36,28,41,33,25,39,32)$ |
| $ 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $11$ | $(23,27,31,35,40,44,26,29,34,37,42)(24,28,32,36,39,43,25,30,33,38,41)$ |
| $ 11, 11, 11, 11 $ | $8$ | $11$ | $( 1,20,15,11, 7, 4,22,17,13, 9, 6)( 2,19,16,12, 8, 3,21,18,14,10, 5) (23,29,35,42,26,31,37,44,27,34,40)(24,30,36,41,25,32,38,43,28,33,39)$ |
| $ 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $11$ | $(23,31,40,26,34,42,27,35,44,29,37)(24,32,39,25,33,41,28,36,43,30,38)$ |
| $ 22, 22 $ | $44$ | $22$ | $( 1,40,17,31,11,23, 6,37,22,29,15,44, 9,35, 4,27,20,42,13,34, 7,26) ( 2,39,18,32,12,24, 5,38,21,30,16,43,10,36, 3,28,19,41,14,33, 8,25)$ |
| $ 22, 22 $ | $44$ | $22$ | $( 1,31, 6,29, 9,27,13,26,17,23,22,44, 4,42, 7,40,11,37,15,35,20,34) ( 2,32, 5,30,10,28,14,25,18,24,21,43, 3,41, 8,39,12,38,16,36,19,33)$ |
| $ 22, 22 $ | $44$ | $22$ | $( 1,23,15,27, 7,31,22,35,13,40, 6,44,20,26,11,29, 4,34,17,37, 9,42) ( 2,24,16,28, 8,32,21,36,14,39, 5,43,19,25,12,30, 3,33,18,38,10,41)$ |
| $ 22, 22 $ | $44$ | $22$ | $( 1,29,13,23, 4,40,15,34, 6,27,17,44, 7,37,20,31, 9,26,22,42,11,35) ( 2,30,14,24, 3,39,16,33, 5,28,18,43, 8,38,19,32,10,25,21,41,12,36)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $22$ | $2$ | $( 1,44)( 2,43)( 3,32)( 4,31)( 5,41)( 6,42)( 7,29)( 8,30)( 9,40)(10,39)(11,27) (12,28)(13,37)(14,38)(15,26)(16,25)(17,35)(18,36)(19,24)(20,23)(21,33)(22,34)$ |
| $ 22, 22 $ | $44$ | $22$ | $( 1,27,22,40,20,29,17,42,15,31,13,44,11,34, 9,23, 7,35, 6,26, 4,37) ( 2,28,21,39,19,30,18,41,16,32,14,43,12,33,10,24, 8,36, 5,25, 3,38)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $121$ | $2$ | $( 3,21)( 4,22)( 5,19)( 6,20)( 7,17)( 8,18)( 9,15)(10,16)(11,13)(12,14)(25,43) (26,44)(27,42)(28,41)(29,40)(30,39)(31,37)(32,38)(33,36)(34,35)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2 $ | $242$ | $4$ | $( 1,24, 9,41)( 2,23,10,42)( 3,34, 8,31)( 4,33, 7,32)( 5,44)( 6,43) (11,30,22,36)(12,29,21,35)(13,39,20,25)(14,40,19,26)(15,28,17,38)(16,27,18,37)$ |
| $ 22, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $44$ | $22$ | $( 1, 5, 9,14,17,21, 4, 8,11,16,20, 2, 6,10,13,18,22, 3, 7,12,15,19)(23,36) (24,35)(25,34)(26,33)(27,32)(28,31)(29,30)(37,43)(38,44)(39,42)(40,41)$ |
| $ 22, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $44$ | $22$ | $( 1,21,20,18,15,14,11,10, 7, 5, 4, 2,22,19,17,16,13,12, 9, 8, 6, 3)(23,28) (24,27)(25,26)(29,43)(30,44)(31,41)(32,42)(33,40)(34,39)(35,38)(36,37)$ |
| $ 22, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $44$ | $22$ | $( 1,16, 7,21,13, 5,20,12, 4,18, 9, 2,15, 8,22,14, 6,19,11, 3,17,10)(23,41) (24,42)(25,40)(26,39)(27,38)(28,37)(29,36)(30,35)(31,33)(32,34)(43,44)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $22$ | $2$ | $( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9,10)(11,12)(13,14)(15,16)(17,18)(19,20)(21,22) (23,38)(24,37)(25,35)(26,36)(27,33)(28,34)(29,32)(30,31)(39,44)(40,43)(41,42)$ |
| $ 22, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $44$ | $22$ | $( 1,12,22,10,20, 8,17, 5,15, 3,13, 2,11,21, 9,19, 7,18, 6,16, 4,14)(23,43) (24,44)(25,42)(26,41)(27,39)(28,40)(29,38)(30,37)(31,36)(32,35)(33,34)$ |
| $ 22, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $44$ | $22$ | $( 1,18,11, 5,22,16, 9, 3,20,14, 7, 2,17,12, 6,21,15,10, 4,19,13, 8)(23,30) (24,29)(25,27)(26,28)(31,43)(32,44)(33,42)(34,41)(35,39)(36,40)(37,38)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $968=2^{3} \cdot 11^{2}$ | magma: Order(G);
| |
| Cyclic: | no | magma: IsCyclic(G);
| |
| Abelian: | no | magma: IsAbelian(G);
| |
| Solvable: | yes | magma: IsSolvable(G);
| |
| Nilpotency class: | not nilpotent | ||
| Label: | 968.36 | magma: IdentifyGroup(G);
|
| Character table: not available. |
magma: CharacterTable(G);