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Magma
magma: G := TransitiveGroup(44, 21);
Group action invariants
| Degree $n$: | $44$ | magma: t, n := TransitiveGroupIdentification(G); n;
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| Transitive number $t$: | $21$ | magma: t, n := TransitiveGroupIdentification(G); t;
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| Group: | $C_{44}:C_{10}$ | ||
| Parity: | $-1$ | magma: IsEven(G);
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| Primitive: | no | magma: IsPrimitive(G);
| magma: NilpotencyClass(G);
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| $\card{\Aut(F/K)}$: | $2$ | magma: Order(Centralizer(SymmetricGroup(n), G));
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| Generators: | (1,29,37,33,13)(2,30,38,34,14)(3,32,40,36,16,4,31,39,35,15)(7,8)(9,27,18,24,44,10,28,17,23,43)(11,25,19,22,41)(12,26,20,21,42), (1,16,25,40,5,17,29,43,11,24,33,3,13,27,37,8,19,31,41,10,22,35)(2,15,26,39,6,18,30,44,12,23,34,4,14,28,38,7,20,32,42,9,21,36) | magma: Generators(G);
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Low degree resolvents
|G/N| Galois groups for stem field(s) $2$: $C_2$ x 3 $4$: $C_2^2$ $5$: $C_5$ $8$: $D_{4}$ $10$: $C_{10}$ x 3 $20$: 20T3 $40$: 20T12 $55$: $C_{11}:C_5$ $110$: 22T5 x 3 $220$: 44T16 Resolvents shown for degrees $\leq 47$
Subfields
Degree 2: $C_2$
Degree 4: $D_{4}$
Degree 11: $C_{11}:C_5$
Degree 22: 22T5
Low degree siblings
44T21Siblings 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 $ | $1$ | $1$ | $()$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ | $11$ | $5$ | $( 5,13,37,22,19)( 6,14,38,21,20)( 7,15,39,23,18)( 8,16,40,24,17) ( 9,28,32,44,36)(10,27,31,43,35)(11,25,29,41,33)(12,26,30,42,34)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ | $11$ | $5$ | $( 5,19,22,37,13)( 6,20,21,38,14)( 7,18,23,39,15)( 8,17,24,40,16) ( 9,36,44,32,28)(10,35,43,31,27)(11,33,41,29,25)(12,34,42,30,26)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ | $11$ | $5$ | $( 5,22,13,19,37)( 6,21,14,20,38)( 7,23,15,18,39)( 8,24,16,17,40) ( 9,44,28,36,32)(10,43,27,35,31)(11,41,25,33,29)(12,42,26,34,30)$ |
| $ 5, 5, 5, 5, 5, 5, 5, 5, 1, 1, 1, 1 $ | $11$ | $5$ | $( 5,37,19,13,22)( 6,38,20,14,21)( 7,39,18,15,23)( 8,40,17,16,24) ( 9,32,36,28,44)(10,31,35,27,43)(11,29,33,25,41)(12,30,34,26,42)$ |
| $ 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 $ | $2$ | $2$ | $( 3, 4)( 7, 8)( 9,10)(15,16)(17,18)(23,24)(27,28)(31,32)(35,36)(39,40)(43,44)$ |
| $ 10, 10, 5, 5, 5, 5, 2, 1, 1 $ | $22$ | $10$ | $( 3, 4)( 5,13,37,22,19)( 6,14,38,21,20)( 7,16,39,24,18, 8,15,40,23,17) ( 9,27,32,43,36,10,28,31,44,35)(11,25,29,41,33)(12,26,30,42,34)$ |
| $ 10, 10, 5, 5, 5, 5, 2, 1, 1 $ | $22$ | $10$ | $( 3, 4)( 5,19,22,37,13)( 6,20,21,38,14)( 7,17,23,40,15, 8,18,24,39,16) ( 9,35,44,31,28,10,36,43,32,27)(11,33,41,29,25)(12,34,42,30,26)$ |
| $ 10, 10, 5, 5, 5, 5, 2, 1, 1 $ | $22$ | $10$ | $( 3, 4)( 5,22,13,19,37)( 6,21,14,20,38)( 7,24,15,17,39, 8,23,16,18,40) ( 9,43,28,35,32,10,44,27,36,31)(11,41,25,33,29)(12,42,26,34,30)$ |
| $ 10, 10, 5, 5, 5, 5, 2, 1, 1 $ | $22$ | $10$ | $( 3, 4)( 5,37,19,13,22)( 6,38,20,14,21)( 7,40,18,16,23, 8,39,17,15,24) ( 9,31,36,27,44,10,32,35,28,43)(11,29,33,25,41)(12,30,34,26,42)$ |
| $ 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 $ | $11$ | $10$ | $( 1, 2)( 3, 4)( 5,14,37,21,19, 6,13,38,22,20)( 7,16,39,24,18, 8,15,40,23,17) ( 9,27,32,43,36,10,28,31,44,35)(11,26,29,42,33,12,25,30,41,34)$ |
| $ 10, 10, 10, 10, 2, 2 $ | $11$ | $10$ | $( 1, 2)( 3, 4)( 5,20,22,38,13, 6,19,21,37,14)( 7,17,23,40,15, 8,18,24,39,16) ( 9,35,44,31,28,10,36,43,32,27)(11,34,41,30,25,12,33,42,29,26)$ |
| $ 10, 10, 10, 10, 2, 2 $ | $11$ | $10$ | $( 1, 2)( 3, 4)( 5,21,13,20,37, 6,22,14,19,38)( 7,24,15,17,39, 8,23,16,18,40) ( 9,43,28,35,32,10,44,27,36,31)(11,42,25,34,29,12,41,26,33,30)$ |
| $ 10, 10, 10, 10, 2, 2 $ | $11$ | $10$ | $( 1, 2)( 3, 4)( 5,38,19,14,22, 6,37,20,13,21)( 7,40,18,16,23, 8,39,17,15,24) ( 9,31,36,27,44,10,32,35,28,43)(11,30,33,26,41,12,29,34,25,42)$ |
| $ 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2 $ | $2$ | $2$ | $( 1, 3)( 2, 4)( 5, 8)( 6, 7)( 9,12)(10,11)(13,16)(14,15)(17,19)(18,20)(21,23) (22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,40)(38,39)(41,43)(42,44)$ |
| $ 10, 10, 10, 10, 2, 2 $ | $22$ | $10$ | $( 1, 3)( 2, 4)( 5,16,37,24,19, 8,13,40,22,17)( 6,15,38,23,20, 7,14,39,21,18) ( 9,26,32,42,36,12,28,30,44,34)(10,25,31,41,35,11,27,29,43,33)$ |
| $ 10, 10, 10, 10, 2, 2 $ | $22$ | $10$ | $( 1, 3)( 2, 4)( 5,17,22,40,13, 8,19,24,37,16)( 6,18,21,39,14, 7,20,23,38,15) ( 9,34,44,30,28,12,36,42,32,26)(10,33,43,29,27,11,35,41,31,25)$ |
| $ 10, 10, 10, 10, 2, 2 $ | $22$ | $10$ | $( 1, 3)( 2, 4)( 5,24,13,17,37, 8,22,16,19,40)( 6,23,14,18,38, 7,21,15,20,39) ( 9,42,28,34,32,12,44,26,36,30)(10,41,27,33,31,11,43,25,35,29)$ |
| $ 10, 10, 10, 10, 2, 2 $ | $22$ | $10$ | $( 1, 3)( 2, 4)( 5,40,19,16,22, 8,37,17,13,24)( 6,39,20,15,21, 7,38,18,14,23) ( 9,30,36,26,44,12,32,34,28,42)(10,29,35,25,43,11,31,33,27,41)$ |
| $ 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4 $ | $2$ | $4$ | $( 1, 3, 2, 4)( 5, 8, 6, 7)( 9,11,10,12)(13,16,14,15)(17,20,18,19)(21,23,22,24) (25,27,26,28)(29,31,30,32)(33,35,34,36)(37,40,38,39)(41,43,42,44)$ |
| $ 20, 20, 4 $ | $22$ | $20$ | $( 1, 3, 2, 4)( 5,16,38,23,19, 8,14,39,22,17, 6,15,37,24,20, 7,13,40,21,18) ( 9,25,31,42,36,11,27,30,44,33,10,26,32,41,35,12,28,29,43,34)$ |
| $ 20, 20, 4 $ | $22$ | $20$ | $( 1, 3, 2, 4)( 5,17,21,39,13, 8,20,23,37,16, 6,18,22,40,14, 7,19,24,38,15) ( 9,33,43,30,28,11,35,42,32,25,10,34,44,29,27,12,36,41,31,26)$ |
| $ 20, 20, 4 $ | $22$ | $20$ | $( 1, 3, 2, 4)( 5,24,14,18,37, 8,21,15,19,40, 6,23,13,17,38, 7,22,16,20,39) ( 9,41,27,34,32,11,43,26,36,29,10,42,28,33,31,12,44,25,35,30)$ |
| $ 20, 20, 4 $ | $22$ | $20$ | $( 1, 3, 2, 4)( 5,40,20,15,22, 8,38,18,13,24, 6,39,19,16,21, 7,37,17,14,23) ( 9,29,35,26,44,11,31,34,28,41,10,30,36,25,43,12,32,33,27,42)$ |
| $ 22, 11, 11 $ | $10$ | $22$ | $( 1, 5,11,13,19,22,25,29,33,37,41)( 2, 6,12,14,20,21,26,30,34,38,42) ( 3, 7,10,15,17,23,27,32,35,39,43, 4, 8, 9,16,18,24,28,31,36,40,44)$ |
| $ 11, 11, 11, 11 $ | $5$ | $11$ | $( 1, 5,11,13,19,22,25,29,33,37,41)( 2, 6,12,14,20,21,26,30,34,38,42) ( 3, 8,10,16,17,24,27,31,35,40,43)( 4, 7, 9,15,18,23,28,32,36,39,44)$ |
| $ 22, 22 $ | $5$ | $22$ | $( 1, 6,11,14,19,21,25,30,33,38,41, 2, 5,12,13,20,22,26,29,34,37,42) ( 3, 7,10,15,17,23,27,32,35,39,43, 4, 8, 9,16,18,24,28,31,36,40,44)$ |
| $ 44 $ | $10$ | $44$ | $( 1, 7,12,16,19,23,26,31,33,39,42, 3, 5, 9,14,17,22,28,30,35,37,44, 2, 8,11, 15,20,24,25,32,34,40,41, 4, 6,10,13,18,21,27,29,36,38,43)$ |
| $ 22, 22 $ | $10$ | $22$ | $( 1, 7,11,15,19,23,25,32,33,39,41, 4, 5, 9,13,18,22,28,29,36,37,44) ( 2, 8,12,16,20,24,26,31,34,40,42, 3, 6,10,14,17,21,27,30,35,38,43)$ |
| $ 44 $ | $10$ | $44$ | $( 1, 9,20,27,33,44, 6,16,22,32,38, 3,11,18,26,35,41, 7,14,24,29,39, 2,10,19, 28,34,43, 5,15,21,31,37, 4,12,17,25,36,42, 8,13,23,30,40)$ |
| $ 22, 22 $ | $10$ | $22$ | $( 1, 9,19,28,33,44, 5,15,22,32,37, 4,11,18,25,36,41, 7,13,23,29,39) ( 2,10,20,27,34,43, 6,16,21,31,38, 3,12,17,26,35,42, 8,14,24,30,40)$ |
| $ 22, 11, 11 $ | $10$ | $22$ | $( 1,11,19,25,33,41, 5,13,22,29,37)( 2,12,20,26,34,42, 6,14,21,30,38) ( 3, 9,17,28,35,44, 8,15,24,32,40, 4,10,18,27,36,43, 7,16,23,31,39)$ |
| $ 11, 11, 11, 11 $ | $5$ | $11$ | $( 1,11,19,25,33,41, 5,13,22,29,37)( 2,12,20,26,34,42, 6,14,21,30,38) ( 3,10,17,27,35,43, 8,16,24,31,40)( 4, 9,18,28,36,44, 7,15,23,32,39)$ |
| $ 22, 22 $ | $5$ | $22$ | $( 1,12,19,26,33,42, 5,14,22,30,37, 2,11,20,25,34,41, 6,13,21,29,38) ( 3, 9,17,28,35,44, 8,15,24,32,40, 4,10,18,27,36,43, 7,16,23,31,39)$ |
magma: ConjugacyClasses(G);
Group invariants
| Order: | $440=2^{3} \cdot 5 \cdot 11$ | 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: | 440.13 | magma: IdentifyGroup(G);
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| Character table: not available. |
magma: CharacterTable(G);