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Magma
magma: G := TransitiveGroup(44, 44);
Group action invariants
| Degree $n$: | $44$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $44$ | 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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| $\card{\Aut(F/K)}$: | $1$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,19,9,15,6,22,3,18,11,14,8,21,5,17,2,13,10,20,7,16,4,12)(23,38,28,35,33,43,27,40,32,37,26,34,31,42,25,39,30,36,24,44,29,41), (1,25)(2,28,11,33)(3,31,10,30)(4,23,9,27)(5,26,8,24)(6,29,7,32)(12,39,20,41)(13,42,19,38)(14,34,18,35)(15,37,17,43)(16,40)(21,44,22,36) | 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: $C_2$ x 3
Degree 4: $C_2^2$
Degree 11: None
Degree 22: None
Low degree siblings
22T10 x 2, 44T39 x 2, 44T40 x 2, 44T41 x 2, 44T44Siblings are shown with degree $\leq 47$
A number field with this Galois group has no arithmetically equivalent fields.
Conjugacy classes
| Label | 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 $ | $4$ | $11$ | $( 1, 9, 6, 3,11, 8, 5, 2,10, 7, 4)(12,19,15,22,18,14,21,17,13,20,16) (23,28,33,27,32,26,31,25,30,24,29)(34,42,39,36,44,41,38,35,43,40,37)$ | |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1, 6,11, 5,10, 4, 9, 3, 8, 2, 7)(12,15,18,21,13,16,19,22,14,17,20) (23,33,32,31,30,29,28,27,26,25,24)(34,39,44,38,43,37,42,36,41,35,40)$ | |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(12,18,13,19,14,20,15,21,16,22,17) (23,32,30,28,26,24,33,31,29,27,25)(34,44,43,42,41,40,39,38,37,36,35)$ | |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)(12,13,14,15,16,17,18,19,20,21,22) (23,30,26,33,29,25,32,28,24,31,27)(34,43,41,39,37,35,44,42,40,38,36)$ | |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1, 8, 4,11, 7, 3,10, 6, 2, 9, 5)(12,14,16,18,20,22,13,15,17,19,21) (23,26,29,32,24,27,30,33,25,28,31)(34,41,37,44,40,36,43,39,35,42,38)$ | |
| $ 11, 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $11$ | $(12,14,16,18,20,22,13,15,17,19,21)(23,32,30,28,26,24,33,31,29,27,25) (34,42,39,36,44,41,38,35,43,40,37)$ | |
| $ 11, 11, 11, 11 $ | $8$ | $11$ | $( 1, 9, 6, 3,11, 8, 5, 2,10, 7, 4)(12,21,19,17,15,13,22,20,18,16,14) (23,26,29,32,24,27,30,33,25,28,31)(34,39,44,38,43,37,42,36,41,35,40)$ | |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1, 6,11, 5,10, 4, 9, 3, 8, 2, 7)(12,17,22,16,21,15,20,14,19,13,18) (23,31,28,25,33,30,27,24,32,29,26)(34,36,38,40,42,44,35,37,39,41,43)$ | |
| $ 11, 11, 11, 11 $ | $8$ | $11$ | $( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)(12,15,18,21,13,16,19,22,14,17,20) (23,28,33,27,32,26,31,25,30,24,29)(34,40,35,41,36,42,37,43,38,44,39)$ | |
| $ 11, 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $11$ | $( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)(12,18,13,19,14,20,15,21,16,22,17) (23,27,31,24,28,32,25,29,33,26,30)$ | |
| $ 11, 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $11$ | $(12,16,20,13,17,21,14,18,22,15,19)(23,30,26,33,29,25,32,28,24,31,27) (34,39,44,38,43,37,42,36,41,35,40)$ | |
| $ 11, 11, 11, 11 $ | $8$ | $11$ | $( 1, 6,11, 5,10, 4, 9, 3, 8, 2, 7)(12,19,15,22,18,14,21,17,13,20,16) (23,29,24,30,25,31,26,32,27,33,28)(34,44,43,42,41,40,39,38,37,36,35)$ | |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(12,22,21,20,19,18,17,16,15,14,13) (23,28,33,27,32,26,31,25,30,24,29)(34,38,42,35,39,43,36,40,44,37,41)$ | |
| $ 11, 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $11$ | $( 1, 7, 2, 8, 3, 9, 4,10, 5,11, 6)(12,13,14,15,16,17,18,19,20,21,22) (23,31,28,25,33,30,27,24,32,29,26)$ | |
| $ 11, 11, 11, 11 $ | $8$ | $11$ | $( 1,11,10, 9, 8, 7, 6, 5, 4, 3, 2)(12,15,18,21,13,16,19,22,14,17,20) (23,24,25,26,27,28,29,30,31,32,33)(34,43,41,39,37,35,44,42,40,38,36)$ | |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)(12,21,19,17,15,13,22,20,18,16,14) (23,33,32,31,30,29,28,27,26,25,24)(34,42,39,36,44,41,38,35,43,40,37)$ | |
| $ 11, 11, 11, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 $ | $8$ | $11$ | $( 1, 2, 3, 4, 5, 6, 7, 8, 9,10,11)(12,14,16,18,20,22,13,15,17,19,21) (23,28,33,27,32,26,31,25,30,24,29)$ | |
| $ 11, 11, 11, 11 $ | $8$ | $11$ | $( 1,10, 8, 6, 4, 2,11, 9, 7, 5, 3)(12,18,13,19,14,20,15,21,16,22,17) (23,25,27,29,31,33,24,26,28,30,32)(34,41,37,44,40,36,43,39,35,42,38)$ | |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1, 8, 4,11, 7, 3,10, 6, 2, 9, 5)(12,19,15,22,18,14,21,17,13,20,16) (23,32,30,28,26,24,33,31,29,27,25)(34,39,44,38,43,37,42,36,41,35,40)$ | |
| $ 11, 11, 11, 11 $ | $4$ | $11$ | $( 1, 4, 7,10, 2, 5, 8,11, 3, 6, 9)(12,15,18,21,13,16,19,22,14,17,20) (23,30,26,33,29,25,32,28,24,31,27)(34,44,43,42,41,40,39,38,37,36,35)$ | |
| $ 22, 22 $ | $44$ | $22$ | $( 1,19, 9,15, 6,22, 3,18,11,14, 8,21, 5,17, 2,13,10,20, 7,16, 4,12) (23,38,28,35,33,43,27,40,32,37,26,34,31,42,25,39,30,36,24,44,29,41)$ | |
| $ 22, 22 $ | $44$ | $22$ | $( 1,15, 3,14, 5,13, 7,12, 9,22,11,21, 2,20, 4,19, 6,18, 8,17,10,16) (23,35,27,37,31,39,24,41,28,43,32,34,25,36,29,38,33,40,26,42,30,44)$ | |
| $ 22, 22 $ | $44$ | $22$ | $( 1,22, 8,13, 4,15,11,17, 7,19, 3,21,10,12, 6,14, 2,16, 9,18, 5,20) (23,43,26,39,29,35,32,42,24,38,27,34,30,41,33,37,25,44,28,40,31,36)$ | |
| $ 22, 22 $ | $44$ | $22$ | $( 1,14, 7,22, 2,19, 8,16, 3,13, 9,21, 4,18,10,15, 5,12,11,20, 6,17) (23,37,24,43,25,38,26,44,27,39,28,34,29,40,30,35,31,41,32,36,33,42)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $22$ | $2$ | $( 1,21)( 2,15)( 3,20)( 4,14)( 5,19)( 6,13)( 7,18)( 8,12)( 9,17)(10,22)(11,16) (23,34)(24,40)(25,35)(26,41)(27,36)(28,42)(29,37)(30,43)(31,38)(32,44)(33,39)$ | |
| $ 22, 22 $ | $44$ | $22$ | $( 1,13,11,19,10,14, 9,20, 8,15, 7,21, 6,16, 5,22, 4,17, 3,12, 2,18) (23,39,32,38,30,37,28,36,26,35,24,34,33,44,31,43,29,42,27,41,25,40)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 $ | $121$ | $2$ | $( 2,11)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,22)(14,21)(15,20)(16,19)(17,18)(23,24) (25,33)(26,32)(27,31)(28,30)(34,36)(37,44)(38,43)(39,42)(40,41)$ | |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 2, 2 $ | $242$ | $4$ | $( 1,25)( 2,28,11,33)( 3,31,10,30)( 4,23, 9,27)( 5,26, 8,24)( 6,29, 7,32) (12,39,20,41)(13,42,19,38)(14,34,18,35)(15,37,17,43)(16,40)(21,44,22,36)$ | |
| $ 22, 22 $ | $44$ | $22$ | $( 1,39, 7,37, 2,35, 8,44, 3,42, 9,40, 4,38,10,36, 5,34,11,43, 6,41) (12,30,18,33,13,25,19,28,14,31,20,23,15,26,21,29,16,32,22,24,17,27)$ | |
| $ 22, 22 $ | $44$ | $22$ | $( 1,36, 2,43, 3,39, 4,35, 5,42, 6,38, 7,34, 8,41, 9,37,10,44,11,40) (12,24,13,30,14,25,15,31,16,26,17,32,18,27,19,33,20,28,21,23,22,29)$ | |
| $ 22, 22 $ | $44$ | $22$ | $( 1,44, 8,38, 4,43,11,37, 7,42, 3,36,10,41, 6,35, 2,40, 9,34, 5,39) (12,29,19,27,15,25,22,23,18,32,14,30,21,28,17,26,13,24,20,33,16,31)$ | |
| $ 22, 22 $ | $44$ | $22$ | $( 1,38, 9,39, 6,40, 3,41,11,42, 8,43, 5,44, 2,34,10,35, 7,36, 4,37) (12,28,20,32,17,25,14,29,22,33,19,26,16,30,13,23,21,27,18,31,15,24)$ | |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $22$ | $2$ | $( 1,42)( 2,38)( 3,34)( 4,41)( 5,37)( 6,44)( 7,40)( 8,36)( 9,43)(10,39)(11,35) (12,25)(13,31)(14,26)(15,32)(16,27)(17,33)(18,28)(19,23)(20,29)(21,24)(22,30)$ | |
| $ 22, 22 $ | $44$ | $22$ | $( 1,41, 3,44, 5,36, 7,39, 9,42,11,34, 2,37, 4,40, 6,43, 8,35,10,38) (12,23,14,24,16,25,18,26,20,27,22,28,13,29,15,30,17,31,19,32,21,33)$ |
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: | 35 x 35 character table |
magma: CharacterTable(G);