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Magma
magma: G := TransitiveGroup(44, 29);
Group action invariants
| Degree $n$: | $44$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $29$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $D_{11}^2$ | ||
| 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: | (1,13)(2,14)(3,12)(4,11)(5,9)(6,10)(7,8)(15,21)(16,22)(17,19)(18,20)(23,37)(24,38)(25,36)(26,35)(27,34)(28,33)(29,32)(30,31)(39,44)(40,43)(41,42), (1,27,12,44,22,37,10,32,19,26,7,41,18,36,5,30,15,24,4,40,14,33)(2,28,11,43,21,38,9,31,20,25,8,42,17,35,6,29,16,23,3,39,13,34) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $22$: $D_{11}$ x 2 $44$: $D_{22}$ x 2 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$ x 3
Degree 4: $C_2^2$
Degree 11: None
Degree 22: 22T9
Low degree siblings
22T9 x 5, 44T29 x 4Siblings 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 $ | $4$ | $11$ | $(23,25,28,29,31,34,35,38,39,42,43)(24,26,27,30,32,33,36,37,40,41,44)$ |
| $ 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $11$ | $(23,28,31,35,39,43,25,29,34,38,42)(24,27,32,36,40,44,26,30,33,37,41)$ |
| $ 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $11$ | $(23,29,35,42,25,31,38,43,28,34,39)(24,30,36,41,26,32,37,44,27,33,40)$ |
| $ 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $11$ | $(23,31,39,25,34,42,28,35,43,29,38)(24,32,40,26,33,41,27,36,44,30,37)$ |
| $ 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $4$ | $11$ | $(23,34,43,31,42,29,39,28,38,25,35)(24,33,44,32,41,30,40,27,37,26,36)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $121$ | $2$ | $( 1, 2)( 3,22)( 4,21)( 5,20)( 6,19)( 7,17)( 8,18)( 9,15)(10,16)(11,14)(12,13) (23,24)(25,44)(26,43)(27,42)(28,41)(29,40)(30,39)(31,37)(32,38)(33,35)(34,36)$ |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1, 4, 5, 7,10,12,14,15,18,19,22)( 2, 3, 6, 8, 9,11,13,16,17,20,21) (23,25,28,29,31,34,35,38,39,42,43)(24,26,27,30,32,33,36,37,40,41,44)$ |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1, 4, 5, 7,10,12,14,15,18,19,22)( 2, 3, 6, 8, 9,11,13,16,17,20,21) (23,28,31,35,39,43,25,29,34,38,42)(24,27,32,36,40,44,26,30,33,37,41)$ |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1, 4, 5, 7,10,12,14,15,18,19,22)( 2, 3, 6, 8, 9,11,13,16,17,20,21) (23,29,35,42,25,31,38,43,28,34,39)(24,30,36,41,26,32,37,44,27,33,40)$ |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1, 4, 5, 7,10,12,14,15,18,19,22)( 2, 3, 6, 8, 9,11,13,16,17,20,21) (23,31,39,25,34,42,28,35,43,29,38)(24,32,40,26,33,41,27,36,44,30,37)$ |
| $ 11, 11, 11, 11 $ | $2$ | $11$ | $( 1, 4, 5, 7,10,12,14,15,18,19,22)( 2, 3, 6, 8, 9,11,13,16,17,20,21) (23,34,43,31,42,29,39,28,38,25,35)(24,33,44,32,41,30,40,27,37,26,36)$ |
| $ 11, 11, 11, 11 $ | $2$ | $11$ | $( 1, 4, 5, 7,10,12,14,15,18,19,22)( 2, 3, 6, 8, 9,11,13,16,17,20,21) (23,35,25,38,28,39,29,42,31,43,34)(24,36,26,37,27,40,30,41,32,44,33)$ |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1, 4, 5, 7,10,12,14,15,18,19,22)( 2, 3, 6, 8, 9,11,13,16,17,20,21) (23,38,29,43,35,28,42,34,25,39,31)(24,37,30,44,36,27,41,33,26,40,32)$ |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1, 4, 5, 7,10,12,14,15,18,19,22)( 2, 3, 6, 8, 9,11,13,16,17,20,21) (23,39,34,28,43,38,31,25,42,35,29)(24,40,33,27,44,37,32,26,41,36,30)$ |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1, 4, 5, 7,10,12,14,15,18,19,22)( 2, 3, 6, 8, 9,11,13,16,17,20,21) (23,42,38,34,29,25,43,39,35,31,28)(24,41,37,33,30,26,44,40,36,32,27)$ |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1, 4, 5, 7,10,12,14,15,18,19,22)( 2, 3, 6, 8, 9,11,13,16,17,20,21) (23,43,42,39,38,35,34,31,29,28,25)(24,44,41,40,37,36,33,32,30,27,26)$ |
| $ 11, 11, 11, 11 $ | $2$ | $11$ | $( 1, 5,10,14,18,22, 4, 7,12,15,19)( 2, 6, 9,13,17,21, 3, 8,11,16,20) (23,25,28,29,31,34,35,38,39,42,43)(24,26,27,30,32,33,36,37,40,41,44)$ |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1, 5,10,14,18,22, 4, 7,12,15,19)( 2, 6, 9,13,17,21, 3, 8,11,16,20) (23,28,31,35,39,43,25,29,34,38,42)(24,27,32,36,40,44,26,30,33,37,41)$ |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1, 5,10,14,18,22, 4, 7,12,15,19)( 2, 6, 9,13,17,21, 3, 8,11,16,20) (23,29,35,42,25,31,38,43,28,34,39)(24,30,36,41,26,32,37,44,27,33,40)$ |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1, 5,10,14,18,22, 4, 7,12,15,19)( 2, 6, 9,13,17,21, 3, 8,11,16,20) (23,31,39,25,34,42,28,35,43,29,38)(24,32,40,26,33,41,27,36,44,30,37)$ |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1, 5,10,14,18,22, 4, 7,12,15,19)( 2, 6, 9,13,17,21, 3, 8,11,16,20) (23,38,29,43,35,28,42,34,25,39,31)(24,37,30,44,36,27,41,33,26,40,32)$ |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1, 5,10,14,18,22, 4, 7,12,15,19)( 2, 6, 9,13,17,21, 3, 8,11,16,20) (23,39,34,28,43,38,31,25,42,35,29)(24,40,33,27,44,37,32,26,41,36,30)$ |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1, 5,10,14,18,22, 4, 7,12,15,19)( 2, 6, 9,13,17,21, 3, 8,11,16,20) (23,42,38,34,29,25,43,39,35,31,28)(24,41,37,33,30,26,44,40,36,32,27)$ |
| $ 11, 11, 11, 11 $ | $2$ | $11$ | $( 1, 5,10,14,18,22, 4, 7,12,15,19)( 2, 6, 9,13,17,21, 3, 8,11,16,20) (23,43,42,39,38,35,34,31,29,28,25)(24,44,41,40,37,36,33,32,30,27,26)$ |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1, 7,14,19, 4,10,15,22, 5,12,18)( 2, 8,13,20, 3, 9,16,21, 6,11,17) (23,28,31,35,39,43,25,29,34,38,42)(24,27,32,36,40,44,26,30,33,37,41)$ |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1, 7,14,19, 4,10,15,22, 5,12,18)( 2, 8,13,20, 3, 9,16,21, 6,11,17) (23,29,35,42,25,31,38,43,28,34,39)(24,30,36,41,26,32,37,44,27,33,40)$ |
| $ 11, 11, 11, 11 $ | $2$ | $11$ | $( 1, 7,14,19, 4,10,15,22, 5,12,18)( 2, 8,13,20, 3, 9,16,21, 6,11,17) (23,31,39,25,34,42,28,35,43,29,38)(24,32,40,26,33,41,27,36,44,30,37)$ |
| $ 11, 11, 11, 11 $ | $2$ | $11$ | $( 1, 7,14,19, 4,10,15,22, 5,12,18)( 2, 8,13,20, 3, 9,16,21, 6,11,17) (23,38,29,43,35,28,42,34,25,39,31)(24,37,30,44,36,27,41,33,26,40,32)$ |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1, 7,14,19, 4,10,15,22, 5,12,18)( 2, 8,13,20, 3, 9,16,21, 6,11,17) (23,39,34,28,43,38,31,25,42,35,29)(24,40,33,27,44,37,32,26,41,36,30)$ |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1, 7,14,19, 4,10,15,22, 5,12,18)( 2, 8,13,20, 3, 9,16,21, 6,11,17) (23,42,38,34,29,25,43,39,35,31,28)(24,41,37,33,30,26,44,40,36,32,27)$ |
| $ 11, 11, 11, 11 $ | $2$ | $11$ | $( 1,10,18, 4,12,19, 5,14,22, 7,15)( 2, 9,17, 3,11,20, 6,13,21, 8,16) (23,28,31,35,39,43,25,29,34,38,42)(24,27,32,36,40,44,26,30,33,37,41)$ |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1,10,18, 4,12,19, 5,14,22, 7,15)( 2, 9,17, 3,11,20, 6,13,21, 8,16) (23,29,35,42,25,31,38,43,28,34,39)(24,30,36,41,26,32,37,44,27,33,40)$ |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1,10,18, 4,12,19, 5,14,22, 7,15)( 2, 9,17, 3,11,20, 6,13,21, 8,16) (23,39,34,28,43,38,31,25,42,35,29)(24,40,33,27,44,37,32,26,41,36,30)$ |
| $ 11, 11, 11, 11 $ | $2$ | $11$ | $( 1,10,18, 4,12,19, 5,14,22, 7,15)( 2, 9,17, 3,11,20, 6,13,21, 8,16) (23,42,38,34,29,25,43,39,35,31,28)(24,41,37,33,30,26,44,40,36,32,27)$ |
| $ 11, 11, 11, 11 $ | $2$ | $11$ | $( 1,12,22,10,19, 7,18, 5,15, 4,14)( 2,11,21, 9,20, 8,17, 6,16, 3,13) (23,29,35,42,25,31,38,43,28,34,39)(24,30,36,41,26,32,37,44,27,33,40)$ |
| $ 11, 11, 11, 11 $ | $2$ | $11$ | $( 1,12,22,10,19, 7,18, 5,15, 4,14)( 2,11,21, 9,20, 8,17, 6,16, 3,13) (23,39,34,28,43,38,31,25,42,35,29)(24,40,33,27,44,37,32,26,41,36,30)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $11$ | $2$ | $( 1,23)( 2,24)( 3,33)( 4,34)( 5,43)( 6,44)( 7,31)( 8,32)( 9,41)(10,42)(11,30) (12,29)(13,40)(14,39)(15,28)(16,27)(17,37)(18,38)(19,25)(20,26)(21,36)(22,35)$ |
| $ 22, 22 $ | $22$ | $22$ | $( 1,23, 4,34, 5,43, 7,31,10,42,12,29,14,39,15,28,18,38,19,25,22,35) ( 2,24, 3,33, 6,44, 8,32, 9,41,11,30,13,40,16,27,17,37,20,26,21,36)$ |
| $ 22, 22 $ | $22$ | $22$ | $( 1,23, 5,43,10,42,14,39,18,38,22,35, 4,34, 7,31,12,29,15,28,19,25) ( 2,24, 6,44, 9,41,13,40,17,37,21,36, 3,33, 8,32,11,30,16,27,20,26)$ |
| $ 22, 22 $ | $22$ | $22$ | $( 1,23, 7,31,14,39,19,25, 4,34,10,42,15,28,22,35, 5,43,12,29,18,38) ( 2,24, 8,32,13,40,20,26, 3,33, 9,41,16,27,21,36, 6,44,11,30,17,37)$ |
| $ 22, 22 $ | $22$ | $22$ | $( 1,23,10,42,18,38, 4,34,12,29,19,25, 5,43,14,39,22,35, 7,31,15,28) ( 2,24, 9,41,17,37, 3,33,11,30,20,26, 6,44,13,40,21,36, 8,32,16,27)$ |
| $ 22, 22 $ | $22$ | $22$ | $( 1,23,12,29,22,35,10,42,19,25, 7,31,18,38, 5,43,15,28, 4,34,14,39) ( 2,24,11,30,21,36, 9,41,20,26, 8,32,17,37, 6,44,16,27, 3,33,13,40)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $11$ | $2$ | $( 1,24)( 2,23)( 3,35)( 4,36)( 5,26)( 6,25)( 7,37)( 8,38)( 9,28)(10,27)(11,39) (12,40)(13,29)(14,30)(15,41)(16,42)(17,31)(18,32)(19,44)(20,43)(21,34)(22,33)$ |
| $ 22, 22 $ | $22$ | $22$ | $( 1,24, 4,36, 5,26, 7,37,10,27,12,40,14,30,15,41,18,32,19,44,22,33) ( 2,23, 3,35, 6,25, 8,38, 9,28,11,39,13,29,16,42,17,31,20,43,21,34)$ |
| $ 22, 22 $ | $22$ | $22$ | $( 1,24, 5,26,10,27,14,30,18,32,22,33, 4,36, 7,37,12,40,15,41,19,44) ( 2,23, 6,25, 9,28,13,29,17,31,21,34, 3,35, 8,38,11,39,16,42,20,43)$ |
| $ 22, 22 $ | $22$ | $22$ | $( 1,24, 7,37,14,30,19,44, 4,36,10,27,15,41,22,33, 5,26,12,40,18,32) ( 2,23, 8,38,13,29,20,43, 3,35, 9,28,16,42,21,34, 6,25,11,39,17,31)$ |
| $ 22, 22 $ | $22$ | $22$ | $( 1,24,10,27,18,32, 4,36,12,40,19,44, 5,26,14,30,22,33, 7,37,15,41) ( 2,23, 9,28,17,31, 3,35,11,39,20,43, 6,25,13,29,21,34, 8,38,16,42)$ |
| $ 22, 22 $ | $22$ | $22$ | $( 1,24,12,40,22,33,10,27,19,44, 7,37,18,32, 5,26,15,41, 4,36,14,30) ( 2,23,11,39,21,34, 9,28,20,43, 8,38,17,31, 6,25,16,42, 3,35,13,29)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $484=2^{2} \cdot 11^{2}$ | 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: | 484.9 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);