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